Libraries

library(readxl)
library(tidyverse)
library(lubridate)
library(zoo)
library(forecast)
library(imputeTS)
library(TSA)
library(tseries)
library(xts)
library(astsa)
library(padr)
library(Metrics)
library(DescTools)
library(lmtest)
library(gridExtra)
library(ggpubr)
library(dLagM)
library(GGally)
setwd("~/Documents/MSc_Data_Science_and_Statistics_Coursework/Dissertation/Updated data and R code")

Data Preprocessing and Exploratory Data Analysis

#Step one: explore reported new cases data near sewage plant with possibility 
#to create a model that represents the whole population? 
#use NC reported cases data?....
reported_cases <- read_excel("New_Cases_Per_10K_updated_in_wwtp.xlsx")
glimpse(reported_cases)
## Rows: 9,815
## Columns: 6
## $ Index                          <chr> "1", "2", "3", "4", "5", "6", "7", "8",…
## $ `Wastewater Treatment Plant`   <chr> "Cary 1", "Cary 1", "Cary 1", "Cary 1",…
## $ County                         <chr> "Wake", "Wake", "Wake", "Wake", "Wake",…
## $ Date                           <chr> "5/25/2022", "5/24/2022", "5/23/2022", …
## $ `Population Served`            <dbl> 84189, 84189, 84189, 84189, 84189, 8418…
## $ `New Cases Per 10,000 Persons` <dbl> NA, NA, 2.02, 3.21, 7.36, 7.84, 9.15, 7…
colnames(reported_cases)[2] <- "wwtp"
colnames(reported_cases)[5] <- "population"
colnames(reported_cases)[6] <- "new_cases_per_10k"
glimpse(reported_cases)
## Rows: 9,815
## Columns: 6
## $ Index             <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "…
## $ wwtp              <chr> "Cary 1", "Cary 1", "Cary 1", "Cary 1", "Cary 1", "C…
## $ County            <chr> "Wake", "Wake", "Wake", "Wake", "Wake", "Wake", "Wak…
## $ Date              <chr> "5/25/2022", "5/24/2022", "5/23/2022", "5/22/2022", …
## $ population        <dbl> 84189, 84189, 84189, 84189, 84189, 84189, 84189, 841…
## $ new_cases_per_10k <dbl> NA, NA, 2.02, 3.21, 7.36, 7.84, 9.15, 7.72, 7.25, 3.…
reported_cases <- reported_cases %>% arrange(mdy(reported_cases$Date))
reported_cases$Date <- mdy(reported_cases$Date)
table(reported_cases$County)
## 
##  Buncombe,Henderson            Carteret             Chatham          Cumberland 
##                 355                 736                 143                 355 
##              Durham             Forsyth            Guilford Halifax,Northampton 
##                 508                 355                 356                 355 
##             Jackson            Mcdowell         Mecklenburg         New Hanover 
##                 508                 357                1375                1011 
##              Onslow              Orange                Pitt            Scotland 
##                  82                 508                 508                 357 
##                Wake              Wilson 
##                1585                 355
png(filename="new_cases_plot.png", res = 500,units = "cm", width = 20, height = 10)
na.omit(reported_cases) %>% ggplot(aes(Date,new_cases_per_10k)) + facet_wrap(~County) + geom_line() + 
  ylab("New COVID-19 cases per 10K") + theme_bw()
dev.off()
## quartz_off_screen 
##                 2
png(filename="log_new_cases_boxplot.png", res = 500,units = "cm", width = 20, height = 12)
na.omit(reported_cases) %>% ggplot(aes(County,log(new_cases_per_10k))) + 
  geom_boxplot() + ylab("Logarithm of new COVID-19 cases per 10k") +
  theme_bw() + theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1)) 
dev.off()
## quartz_off_screen 
##                 2
#Wake: New Cases
wake_reported_cases <- subset(reported_cases,County=="Wake")
glimpse(wake_reported_cases)
## Rows: 1,585
## Columns: 6
## $ Index             <chr> "6,611", "6,610", "6,609", "6,608", "6,607", "6,606"…
## $ wwtp              <chr> "Raleigh", "Raleigh", "Raleigh", "Raleigh", "Raleigh…
## $ County            <chr> "Wake", "Wake", "Wake", "Wake", "Wake", "Wake", "Wak…
## $ Date              <date> 2021-01-03, 2021-01-04, 2021-01-05, 2021-01-06, 202…
## $ population        <dbl> 550000, 550000, 550000, 550000, 550000, 550000, 5500…
## $ new_cases_per_10k <dbl> 8.76, 3.15, 13.69, 10.56, 11.00, 11.16, 8.64, 5.29, …
summary(wake_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1585        Length:1585        Length:1585        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-11-08  
##  Mode  :character   Mode  :character   Mode  :character   Median :2022-01-13  
##                                                           Mean   :2021-12-21  
##                                                           3rd Qu.:2022-03-20  
##                                                           Max.   :2022-05-25  
##                                                                               
##    population     new_cases_per_10k
##  Min.   :  7776   Min.   :  0.040  
##  1st Qu.: 30655   1st Qu.:  1.310  
##  Median : 75886   Median :  2.570  
##  Mean   :212196   Mean   :  7.511  
##  3rd Qu.:550000   3rd Qu.:  6.430  
##  Max.   :550000   Max.   :109.310  
##                   NA's   :75
which(is.na(wake_reported_cases$`New Cases Per 10,000 Persons`))
## integer(0)
wake_reported_cases %>% slice_min(new_cases_per_10k)
## # A tibble: 1 × 6
##   Index wwtp    County Date       population new_cases_per_10k
##   <chr> <chr>   <chr>  <date>          <dbl>             <dbl>
## 1 6,428 Raleigh Wake   2021-07-05     550000              0.04
wake_reported_cases %>% slice_max(new_cases_per_10k)
## # A tibble: 1 × 6
##   Index wwtp      County Date       population new_cases_per_10k
##   <chr> <chr>     <chr>  <date>          <dbl>             <dbl>
## 1 6,983 Raleigh 3 Wake   2022-01-05       7776              109.
wake_reported_cases$new_cases_per_10k <- LOCF(wake_reported_cases$new_cases_per_10k)
summary(wake_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1585        Length:1585        Length:1585        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-11-08  
##  Mode  :character   Mode  :character   Mode  :character   Median :2022-01-13  
##                                                           Mean   :2021-12-21  
##                                                           3rd Qu.:2022-03-20  
##                                                           Max.   :2022-05-25  
##    population     new_cases_per_10k
##  Min.   :  7776   Min.   :  0.040  
##  1st Qu.: 30655   1st Qu.:  1.160  
##  Median : 75886   Median :  2.570  
##  Mean   :212196   Mean   :  7.217  
##  3rd Qu.:550000   3rd Qu.:  6.200  
##  Max.   :550000   Max.   :109.310
df_wake <- wake_reported_cases %>% group_by(Date) %>% summarise(mean_new_cases=mean(new_cases_per_10k))
summary(df_wake)
##       Date            mean_new_cases  
##  Min.   :2021-01-03   Min.   : 0.040  
##  1st Qu.:2021-05-09   1st Qu.: 1.175  
##  Median :2021-09-13   Median : 2.390  
##  Mean   :2021-09-13   Mean   : 5.049  
##  3rd Qu.:2022-01-18   3rd Qu.: 4.559  
##  Max.   :2022-05-25   Max.   :65.640
#Wake: wastewater viral gene copies

wastewater_data <- read_excel("wastewater_data_updated.xlsx")
glimpse(wastewater_data)
## Rows: 2,554
## Columns: 7
## $ Index                          <chr> "1", "2", "3", "4", "5", "6", "7", "8",…
## $ `Wastewater Treatment Plant`   <chr> "Cary 1", "Cary 1", "Cary 1", "Cary 1",…
## $ County                         <chr> "Wake", "Wake", "Wake", "Wake", "Wake",…
## $ Date                           <chr> "5/24/2022", "5/19/2022", "5/17/2022", …
## $ `Population Served`            <dbl> 84189, 84189, 84189, 84189, 84189, 8418…
## $ `Viral Gene Copies Per Person` <dbl> 27231850.1, 41893904.5, 24733220.3, 216…
## $ `Viral Gene Copies/L`          <dbl> 72706.915, 138858.315, 71905.615, 71604…
wastewater_data <- wastewater_data%>% arrange(mdy(wastewater_data$Date))
wastewater_data$Date <- mdy(wastewater_data$Date)
colnames(wastewater_data)[2]<-"wwtp"
colnames(wastewater_data)[5]<-"population"
colnames(wastewater_data)[6]<-"viral_gene_copies_per_person"
colnames(wastewater_data)[7]<-"viral_gene_copies/L"

summary(wastewater_data) #no missing values
##     Index               wwtp              County               Date           
##  Length:2554        Length:2554        Length:2554        Min.   :2021-01-04  
##  Class :character   Class :character   Class :character   1st Qu.:2021-07-10  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-11-06  
##                                                           Mean   :2021-10-23  
##                                                           3rd Qu.:2022-02-23  
##                                                           Max.   :2022-05-25  
##    population     viral_gene_copies_per_person viral_gene_copies/L
##  Min.   :  3500   Min.   :    39617            Min.   :      0    
##  1st Qu.: 15527   1st Qu.:  1219550            1st Qu.:   3536    
##  Median : 74331   Median :  5043606            Median :  13325    
##  Mean   : 99804   Mean   : 14331526            Mean   :  38399    
##  3rd Qu.:120000   3rd Qu.: 14409756            3rd Qu.:  37871    
##  Max.   :550000   Max.   :861501758            Max.   :2646342
png(filename="viral_gene_plot.png", res = 500,units = "cm", width = 20, height = 10)
wastewater_data %>% ggplot(aes(Date,viral_gene_copies_per_person)) + geom_line() + 
  facet_wrap(~ County) + theme_bw() + ylab("Viral gene copies per person")
dev.off()
## quartz_off_screen 
##                 2
png(filename="log_viral_gene_boxplot.png", res = 500,units = "cm", width = 20, height = 10)
wastewater_data %>% ggplot(aes(County,log(viral_gene_copies_per_person))) + geom_boxplot() +
  theme_bw() + theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1)) + 
    ylab("log(Viral gene copies per person)")
dev.off()
## quartz_off_screen 
##                 2
wake_wastewater_data <- wastewater_data %>% dplyr::filter(County=='Wake')
summary(wake_wastewater_data)
##     Index               wwtp              County               Date           
##  Length:389         Length:389         Length:389         Min.   :2021-01-06  
##  Class :character   Class :character   Class :character   1st Qu.:2021-11-18  
##  Mode  :character   Mode  :character   Mode  :character   Median :2022-01-28  
##                                                           Mean   :2021-12-28  
##                                                           3rd Qu.:2022-03-29  
##                                                           Max.   :2022-05-25  
##    population     viral_gene_copies_per_person viral_gene_copies/L
##  Min.   :  7776   Min.   :    90179            Min.   :      0    
##  1st Qu.: 30655   1st Qu.:  2263004            1st Qu.:   7030    
##  Median : 75886   Median :  6776605            Median :  20494    
##  Mean   :218051   Mean   : 20720759            Mean   :  61134    
##  3rd Qu.:550000   3rd Qu.: 18414486            3rd Qu.:  59674    
##  Max.   :550000   Max.   :861501758            Max.   :2646342
wake_wastewater_data %>% slice_min(viral_gene_copies_per_person)
## # A tibble: 1 × 7
##   Index wwtp    County Date       population viral_gene_copies… `viral_gene_co…`
##   <chr> <chr>   <chr>  <date>          <dbl>              <dbl>            <dbl>
## 1 1,696 Raleigh Wake   2021-05-08     550000             90179.                0
wake_wastewater_data %>% slice_max(viral_gene_copies_per_person)
## # A tibble: 1 × 7
##   Index wwtp    County Date       population viral_gene_copies… `viral_gene_co…`
##   <chr> <chr>   <chr>  <date>          <dbl>              <dbl>            <dbl>
## 1 1,638 Raleigh Wake   2022-01-14     550000         861501758.         2646342.
png(filename="viral_gene_plot_wake.png", res = 500,units = "cm", width = 20, height = 10)
wake_wastewater_data %>% ggplot(aes(Date,viral_gene_copies_per_person)) + geom_line() + 
  facet_wrap(~ wwtp) + theme_bw(base_size = 16) + ylab("Viral gene copies per person")
dev.off()
## quartz_off_screen 
##                 2
png(filename="log_viral_gene_plot_wake.png", res = 500,units = "cm", width = 20, height = 10)
wake_wastewater_data %>% ggplot(aes(wwtp,log(viral_gene_copies_per_person))) + 
  geom_boxplot() + theme_bw(base_size = 16) + 
  ylab("Log(viral gene copies per person)") + 
  xlab("Wastewater treatment plants")
dev.off()
## quartz_off_screen 
##                 2
average_wake_wastewater <- wake_wastewater_data %>% group_by(Date) %>% 
  summarise(mean_viral_gene_copies_per_person = mean(viral_gene_copies_per_person))

average_wake_wastewater
## # A tibble: 188 × 2
##    Date       mean_viral_gene_copies_per_person
##    <date>                                 <dbl>
##  1 2021-01-06                         24940866.
##  2 2021-01-09                         13086649.
##  3 2021-01-13                         20010814.
##  4 2021-01-16                         15455316.
##  5 2021-01-20                          8641657.
##  6 2021-01-23                          9986540.
##  7 2021-01-27                         11114735.
##  8 2021-01-30                          2670705.
##  9 2021-02-03                          5430043.
## 10 2021-02-13                          7822743.
## # … with 178 more rows
full_average_wake_wastewater <- pad(average_wake_wastewater,start_val = as.Date('2021-01-04'),
                                    end_val = as.Date('2022-05-25'))
full_average_wake_wastewater <- full_average_wake_wastewater %>% 
  mutate(full_viral_gene_copies_per_person = na_locf(mean_viral_gene_copies_per_person))
full_average_wake_wastewater %>% ggplot(aes(Date,full_viral_gene_copies_per_person)) + geom_line()

full_average_wake_wastewater <- full_average_wake_wastewater %>% select(Date,full_viral_gene_copies_per_person)
summary(full_average_wake_wastewater)
##       Date            full_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   :    90179                
##  1st Qu.:2021-05-10   1st Qu.:  1587873                
##  Median :2021-09-14   Median :  5055290                
##  Mean   :2021-09-14   Mean   : 20579092                
##  3rd Qu.:2022-01-18   3rd Qu.: 12713191                
##  Max.   :2022-05-25   Max.   :861501758
#Wake: merged data

full_cases_wastewater_data <- merge(df_wake[-1,],full_average_wake_wastewater,by="Date")
summary(full_cases_wastewater_data )
##       Date            mean_new_cases   full_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   : 0.040   Min.   :    90179                
##  1st Qu.:2021-05-10   1st Qu.: 1.170   1st Qu.:  1587873                
##  Median :2021-09-14   Median : 2.380   Median :  5055290                
##  Mean   :2021-09-14   Mean   : 5.042   Mean   : 20579092                
##  3rd Qu.:2022-01-18   3rd Qu.: 4.554   3rd Qu.: 12713191                
##  Max.   :2022-05-25   Max.   :65.640   Max.   :861501758
which.min(full_cases_wastewater_data$mean_new_cases)
## [1] 183
full_cases_wastewater_data[183,]
##           Date mean_new_cases full_viral_gene_copies_per_person
## 183 2021-07-05           0.04                           1587873
which.min(full_cases_wastewater_data$full_viral_gene_copies_per_person)
## [1] 125
full_cases_wastewater_data[125,] #minimum covid-19 prevalence within same season 
##           Date mean_new_cases full_viral_gene_copies_per_person
## 125 2021-05-08           1.33                          90179.23
which.max(full_cases_wastewater_data$mean_new_cases)
## [1] 367
full_cases_wastewater_data[367,]
##           Date mean_new_cases full_viral_gene_copies_per_person
## 367 2022-01-05          65.64                          46312926
which.max(full_cases_wastewater_data$full_viral_gene_copies_per_person)
## [1] 376
full_cases_wastewater_data[376,] #maximum covid-19 prevalence within same month
##           Date mean_new_cases full_viral_gene_copies_per_person
## 376 2022-01-14       47.66167                         861501758
#Mecklenburg: covid_cases

mecklenburg_reported_cases <- subset(reported_cases,County=="Mecklenburg")
glimpse(mecklenburg_reported_cases)
## Rows: 1,375
## Columns: 6
## $ Index             <chr> "1,631", "2,139", "1,630", "2,138", "1,629", "2,137"…
## $ wwtp              <chr> "Charlotte 1", "Charlotte 2", "Charlotte 1", "Charlo…
## $ County            <chr> "Mecklenburg", "Mecklenburg", "Mecklenburg", "Meckle…
## $ Date              <date> 2021-01-03, 2021-01-03, 2021-01-04, 2021-01-04, 202…
## $ population        <dbl> 68685, 182501, 68685, 182501, 68685, 182501, 68685, …
## $ new_cases_per_10k <dbl> 7.86, 5.53, 6.99, 4.11, 13.25, 10.47, 9.46, 7.23, 8.…
summary(mecklenburg_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1375        Length:1375        Length:1375        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-06-16  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-10-08  
##                                                           Mean   :2021-10-02  
##                                                           3rd Qu.:2022-01-31  
##                                                           Max.   :2022-05-25  
##                                                                               
##    population     new_cases_per_10k
##  Min.   : 68685   Min.   : 0.110   
##  1st Qu.: 68685   1st Qu.: 0.990   
##  Median :120000   Median : 2.040   
##  Mean   :124133   Mean   : 3.982   
##  3rd Qu.:182501   3rd Qu.: 4.330   
##  Max.   :182501   Max.   :47.580   
##                   NA's   :10
mecklenburg_reported_cases$new_cases_per_10k <- LOCF(mecklenburg_reported_cases$new_cases_per_10k)
df_mecklenburg <- mecklenburg_reported_cases %>% group_by(Date) %>% summarise(mean_new_cases = 
                                                                                mean(new_cases_per_10k))
summary(df_mecklenburg[-1,])
##       Date            mean_new_cases  
##  Min.   :2021-01-04   Min.   : 0.190  
##  1st Qu.:2021-05-10   1st Qu.: 1.048  
##  Median :2021-09-14   Median : 2.070  
##  Mean   :2021-09-14   Mean   : 3.816  
##  3rd Qu.:2022-01-18   3rd Qu.: 4.085  
##  Max.   :2022-05-25   Max.   :38.407
#Mecklenburg: wastewater viral gene copies

mecklenburg_wastewater_data <- wastewater_data %>% dplyr::filter(County=='Mecklenburg')
summary(mecklenburg_wastewater_data)
##     Index               wwtp              County               Date           
##  Length:357         Length:357         Length:357         Min.   :2021-01-04  
##  Class :character   Class :character   Class :character   1st Qu.:2021-06-11  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-10-02  
##                                                           Mean   :2021-09-29  
##                                                           3rd Qu.:2022-02-05  
##                                                           Max.   :2022-05-24  
##    population     viral_gene_copies_per_person viral_gene_copies/L
##  Min.   : 68685   Min.   :    72703            Min.   :      0    
##  1st Qu.: 68685   1st Qu.:  3178201            1st Qu.:  10238    
##  Median :120000   Median :  7729720            Median :  24749    
##  Mean   :124630   Mean   : 16956243            Mean   :  54897    
##  3rd Qu.:182501   3rd Qu.: 19772480            3rd Qu.:  62500    
##  Max.   :182501   Max.   :379065946            Max.   :1101437
png(filename="viral_gene_plot_meck.png", res = 500,units = "cm", width = 20, height = 10)
mecklenburg_wastewater_data%>% ggplot(aes(Date,viral_gene_copies_per_person)) + geom_line() + 
  facet_wrap(~ wwtp) + theme_bw(base_size = 16) + ylab("Viral gene copies per person")
dev.off()
## quartz_off_screen 
##                 2
png(filename="log_viral_gene_plot_meck.png", res = 500,units = "cm", width = 20, height = 10)
mecklenburg_wastewater_data %>% ggplot(aes(wwtp,log(viral_gene_copies_per_person))) +
  geom_boxplot()+ theme_bw(base_size = 16) + 
  ylab("Log(viral gene copies per person)") +xlab("Wastewater treatment plants")
dev.off()
## quartz_off_screen 
##                 2
average_meck_wastewater <- mecklenburg_wastewater_data %>% group_by(Date) %>% 
  summarise(mean_viral_gene_copies_per_person = mean(viral_gene_copies_per_person))

average_meck_wastewater
## # A tibble: 152 × 2
##    Date       mean_viral_gene_copies_per_person
##    <date>                                 <dbl>
##  1 2021-01-04                         48863073.
##  2 2021-01-05                         16458455.
##  3 2021-01-06                         28467455.
##  4 2021-01-11                         11486856.
##  5 2021-01-12                         19831661.
##  6 2021-01-13                         22175968.
##  7 2021-01-20                         13991321.
##  8 2021-01-25                         18933735.
##  9 2021-01-27                         12279871.
## 10 2021-02-01                          4781264.
## # … with 142 more rows
full_average_meck_wastewater <- pad(average_meck_wastewater,start_val = as.Date('2021-01-04'),
                                    end_val = as.Date('2022-05-25'))
full_average_meck_wastewater <- full_average_meck_wastewater %>% 
  mutate(full_viral_gene_copies_per_person = na_locf(mean_viral_gene_copies_per_person))

full_average_meck_wastewater%>% ggplot(aes(Date,full_viral_gene_copies_per_person)) + 
  geom_line()

full_average_wake_wastewater <- full_average_meck_wastewater %>% select(Date,full_viral_gene_copies_per_person)
summary(full_average_meck_wastewater)
##       Date            mean_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   :   283193                
##  1st Qu.:2021-05-10   1st Qu.:  3821535                
##  Median :2021-09-14   Median :  7985329                
##  Mean   :2021-09-14   Mean   : 18052981                
##  3rd Qu.:2022-01-18   3rd Qu.: 20257270                
##  Max.   :2022-05-25   Max.   :379065946                
##                       NA's   :355                      
##  full_viral_gene_copies_per_person
##  Min.   :   283193                
##  1st Qu.:  3818869                
##  Median :  8040114                
##  Mean   : 16676303                
##  3rd Qu.: 19388197                
##  Max.   :379065946                
## 
#Mecklenburg: merged data

full_cases_wastewater_data_meck <- merge(df_mecklenburg,full_average_meck_wastewater,by="Date")
summary(full_cases_wastewater_data_meck)
##       Date            mean_new_cases   mean_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   : 0.190   Min.   :   283193                
##  1st Qu.:2021-05-10   1st Qu.: 1.048   1st Qu.:  3821535                
##  Median :2021-09-14   Median : 2.070   Median :  7985329                
##  Mean   :2021-09-14   Mean   : 3.816   Mean   : 18052981                
##  3rd Qu.:2022-01-18   3rd Qu.: 4.085   3rd Qu.: 20257270                
##  Max.   :2022-05-25   Max.   :38.407   Max.   :379065946                
##                                        NA's   :355                      
##  full_viral_gene_copies_per_person
##  Min.   :   283193                
##  1st Qu.:  3818869                
##  Median :  8040114                
##  Mean   : 16676303                
##  3rd Qu.: 19388197                
##  Max.   :379065946                
## 
which.min(full_cases_wastewater_data_meck$mean_new_cases)
## [1] 155
full_cases_wastewater_data_meck[155,]
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 155 2021-06-07           0.19                           2105482
##     full_viral_gene_copies_per_person
## 155                           2105482
which.min(full_cases_wastewater_data_meck$full_viral_gene_copies_per_person)
## [1] 66
full_cases_wastewater_data_meck[66,] #minimum covid-19 prevalence not in the same period.....
##          Date mean_new_cases mean_viral_gene_copies_per_person
## 66 2021-03-10          1.695                          283192.9
##    full_viral_gene_copies_per_person
## 66                          283192.9
which.max(full_cases_wastewater_data_meck$mean_new_cases)
## [1] 368
full_cases_wastewater_data_meck[368,]
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 368 2022-01-06       38.40667                                NA
##     full_viral_gene_copies_per_person
## 368                         125299582
which.max(full_cases_wastewater_data_meck$full_viral_gene_copies_per_person)
## [1] 369
full_cases_wastewater_data_meck[369,] #maximum covid-19 prevalence within same month, reflecting the peak
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 369 2022-01-07       33.34667                         379065946
##     full_viral_gene_copies_per_person
## 369                         379065946
#New Hanover: covid cases

new_hanover_reported_cases <- subset(reported_cases,County=="New Hanover")
glimpse(new_hanover_reported_cases)
## Rows: 1,011
## Columns: 6
## $ Index             <chr> "9,460", "9,459", "9,458", "9,457", "9,456", "5,726"…
## $ wwtp              <chr> "Wilmington City", "Wilmington City", "Wilmington Ci…
## $ County            <chr> "New Hanover", "New Hanover", "New Hanover", "New Ha…
## $ Date              <date> 2021-01-03, 2021-01-04, 2021-01-05, 2021-01-06, 202…
## $ population        <dbl> 58361, 58361, 58361, 58361, 58361, 67743, 58361, 677…
## $ new_cases_per_10k <dbl> 4.63, 3.77, 9.42, 8.22, 7.20, 6.05, 5.48, 6.64, 4.80…
summary(new_hanover_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1011        Length:1011        Length:1011        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-05-11  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-09-15  
##                                                           Mean   :2021-09-14  
##                                                           3rd Qu.:2022-01-19  
##                                                           Max.   :2022-05-25  
##                                                                               
##    population    new_cases_per_10k
##  Min.   :58361   Min.   : 0.300   
##  1st Qu.:58361   1st Qu.: 0.340   
##  Median :58361   Median : 1.710   
##  Mean   :63029   Mean   : 3.529   
##  3rd Qu.:67743   3rd Qu.: 3.600   
##  Max.   :67743   Max.   :48.490   
##                  NA's   :37
new_hanover_reported_cases$new_cases_per_10k <- LOCF(new_hanover_reported_cases$new_cases_per_10k)
summary(new_hanover_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1011        Length:1011        Length:1011        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-05-11  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-09-15  
##                                                           Mean   :2021-09-14  
##                                                           3rd Qu.:2022-01-19  
##                                                           Max.   :2022-05-25  
##    population    new_cases_per_10k
##  Min.   :58361   Min.   : 0.300   
##  1st Qu.:58361   1st Qu.: 0.340   
##  Median :58361   Median : 1.540   
##  Mean   :63029   Mean   : 3.412   
##  3rd Qu.:67743   3rd Qu.: 3.430   
##  Max.   :67743   Max.   :48.490
df_new_hanover <- new_hanover_reported_cases %>% group_by(Date) %>% 
  summarise(mean_new_cases = mean(new_cases_per_10k))
summary(df_new_hanover[-1,])
##       Date            mean_new_cases  
##  Min.   :2021-01-04   Min.   : 0.300  
##  1st Qu.:2021-05-10   1st Qu.: 0.580  
##  Median :2021-09-14   Median : 1.680  
##  Mean   :2021-09-14   Mean   : 3.426  
##  3rd Qu.:2022-01-18   3rd Qu.: 3.580  
##  Max.   :2022-05-25   Max.   :37.870
#New Hanover: wastewater viral gene copies per person 

new_hanover_wastewater_data <- wastewater_data %>% dplyr::filter(County=='New Hanover')

png(filename="viral_gene_plot_hanover.png", res = 500,units = "cm", width = 20, height = 10)
new_hanover_wastewater_data%>% ggplot(aes(Date,viral_gene_copies_per_person)) + geom_line() + 
  facet_wrap(~ wwtp) + theme_bw(base_size = 16) + ylab("Viral gene copies per person")
dev.off()
## quartz_off_screen 
##                 2
png(filename="log_viral_gene_plot_hanover.png", res = 500,units = "cm", width = 20, height = 10)
new_hanover_wastewater_data %>% ggplot(aes(wwtp,log(viral_gene_copies_per_person))) +
  geom_boxplot() + theme_bw(base_size = 16) + xlab("Log(viral gene copies per person)")
dev.off()
## quartz_off_screen 
##                 2
average_hanover_wastewater <- new_hanover_wastewater_data %>% group_by(Date) %>% 
  summarise(mean_viral_gene_copies_per_person = mean(viral_gene_copies_per_person))

full_average_hanover_wastewater <- pad(average_hanover_wastewater,start_val = as.Date('2021-01-04'),
                                       end_val = as.Date('2022-05-25'))
full_average_hanover_wastewater <- full_average_hanover_wastewater %>% 
  mutate(full_viral_gene_copies_per_person = na_locf(mean_viral_gene_copies_per_person))
average_hanover_wastewater %>% ggplot(aes(Date,mean_viral_gene_copies_per_person)) + geom_line()

full_average_hanover_wastewater <- full_average_hanover_wastewater %>% select(Date,full_viral_gene_copies_per_person)
summary(full_average_hanover_wastewater)
##       Date            full_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   :   74540                 
##  1st Qu.:2021-05-10   1st Qu.:  834407                 
##  Median :2021-09-14   Median : 3676553                 
##  Mean   :2021-09-14   Mean   : 8561704                 
##  3rd Qu.:2022-01-18   3rd Qu.: 8560494                 
##  Max.   :2022-05-25   Max.   :74023842
#New Hanover: Merged Data

full_cases_wastewater_data_hanover <- merge(df_new_hanover[-1,],full_average_hanover_wastewater,by="Date")
summary(full_cases_wastewater_data_hanover)
##       Date            mean_new_cases   full_viral_gene_copies_per_person
##  Min.   :2021-01-04   Min.   : 0.300   Min.   :   74540                 
##  1st Qu.:2021-05-10   1st Qu.: 0.580   1st Qu.:  834407                 
##  Median :2021-09-14   Median : 1.680   Median : 3676553                 
##  Mean   :2021-09-14   Mean   : 3.426   Mean   : 8561704                 
##  3rd Qu.:2022-01-18   3rd Qu.: 3.580   3rd Qu.: 8560494                 
##  Max.   :2022-05-25   Max.   :37.870   Max.   :74023842
which.min(full_cases_wastewater_data_hanover$mean_new_cases)
## [1] 141
full_cases_wastewater_data_meck[141,]
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 141 2021-05-24          0.365                          306292.8
##     full_viral_gene_copies_per_person
## 141                          306292.8
which.min(full_cases_wastewater_data_hanover$full_viral_gene_copies_per_person)
## [1] 124
full_cases_wastewater_data_meck[124,] #minimum covid-19 prevalence in the same period.....
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 124 2021-05-07          0.805                                NA
##     full_viral_gene_copies_per_person
## 124                           5860626
which.max(full_cases_wastewater_data_hanover$mean_new_cases)
## [1] 382
full_cases_wastewater_data_meck[382,]
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 382 2022-01-20       26.45333                                NA
##     full_viral_gene_copies_per_person
## 382                          94292358
which.max(full_cases_wastewater_data_hanover$full_viral_gene_copies_per_person)
## [1] 384
full_cases_wastewater_data_meck[384,] #maximum covid-19 prevalence within same month, reflecting the peak
##           Date mean_new_cases mean_viral_gene_copies_per_person
## 384 2022-01-22       14.81333                          71015872
##     full_viral_gene_copies_per_person
## 384                          71015872
#plot of wake, meck and new hanover#

png(filename = "cases_analysis.png", units = "cm", res = 700,
    width = 20, height = 10)
full_cases_wastewater_data %>% ggplot(aes(Date,mean_new_cases)) + 
  geom_line(aes(color="Wake")) +
  geom_line(data = full_cases_wastewater_data_meck,
            aes(Date,mean_new_cases, color="Mecklenburg")) +
  geom_line(data = full_cases_wastewater_data_hanover,
            aes(Date,mean_new_cases, color="New Hanover")) +
  scale_colour_manual(values=c("Wake"="mediumseagreen", "Mecklenburg"="mediumpurple1", 
                               "New Hanover"="maroon1"), 
                      labels=c("Wake", "Mecklenburg", "New Hanover")) + theme_bw() +
  ylab("New COVID-19 cases") + theme(legend.position = "bottom")
dev.off()
## quartz_off_screen 
##                 2
png(filename = "wastewater_analysis.png", units = "cm", res = 700,
    width = 20, height = 10)
full_cases_wastewater_data %>% ggplot(aes(Date,full_viral_gene_copies_per_person)) + 
  geom_line(aes(color="Wake")) +
  geom_line(data = full_cases_wastewater_data_meck,
            aes(Date,full_viral_gene_copies_per_person, color="Mecklenburg")) +
  geom_line(data = full_cases_wastewater_data_hanover,
            aes(Date,full_viral_gene_copies_per_person, color="New Hanover")) +
  scale_colour_manual(values=c("Wake"="mediumseagreen", "Mecklenburg"="mediumpurple1", 
                               "New Hanover"="maroon1"), 
                      labels=c("Wake", "Mecklenburg", "New Hanover")) + theme_bw() +
  ylab("Viral gene copies per person") + theme(legend.position = "bottom")
dev.off()
## quartz_off_screen 
##                 2
#Weather properties that affect wastewater#

#Wake weather: data preprocessing#
wake_weather <- read.csv("wake_weather.csv")
glimpse(wake_weather)
## Rows: 32,854
## Columns: 8
## $ STATION <chr> "US1NCWK0022", "US1NCWK0022", "US1NCWK0022", "US1NCWK0022", "U…
## $ NAME    <chr> "APEX 6.1 ESE, NC US", "APEX 6.1 ESE, NC US", "APEX 6.1 ESE, N…
## $ DATE    <chr> "2021-01-04", "2021-01-05", "2021-01-06", "2021-01-07", "2021-…
## $ PRCP    <dbl> 0.03, 0.00, 0.00, 0.00, 0.26, 0.45, 0.00, 0.00, 0.15, 0.05, 0.…
## $ TAVG    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TMAX    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TMIN    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TOBS    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
sum(is.na(wake_weather$PRCP)) #757 NA, assume no rain...
## [1] 757
#which(is.na(wake_weather$PRCP))
wake_weather <- 
  wake_weather %>% mutate(PRCP = replace_na(PRCP,mean(PRCP,na.rm=TRUE)))

wake_weather_prcp <- wake_weather %>% group_by(DATE) %>% summarise(mean_precipation = mean(PRCP))

sum(is.na(wake_weather$TAVG))
## [1] 32347
sum(is.na(wake_weather$TMIN))
## [1] 29389
sum(is.na(wake_weather$TMAX))
## [1] 29371
sum(is.na(wake_weather$TOBS))
## [1] 29886
#temperature is the same in all towns within Wake
full_tavg_data <- wake_weather[complete.cases(wake_weather$TAVG),]  
table(full_tavg_data$STATION) #only one stations 
## 
## USW00013722 
##         507
wake_temp <- full_tavg_data %>% select(DATE,TAVG)

wake_weather <- merge(wake_weather_prcp,wake_temp,order.by=DATE)
summary(wake_weather)
##      DATE           mean_precipation        TAVG      
##  Length:507         Min.   :0.000000   Min.   :25.00  
##  Class :character   1st Qu.:0.002439   1st Qu.:46.50  
##  Mode  :character   Median :0.009689   Median :61.00  
##                     Mean   :0.130198   Mean   :59.61  
##                     3rd Qu.:0.092545   3rd Qu.:72.00  
##                     Max.   :2.855180   Max.   :86.00
wake_weather$DATE <- as.Date(wake_weather$DATE)
summary(wake_weather)
##       DATE            mean_precipation        TAVG      
##  Min.   :2021-01-04   Min.   :0.000000   Min.   :25.00  
##  1st Qu.:2021-05-10   1st Qu.:0.002439   1st Qu.:46.50  
##  Median :2021-09-14   Median :0.009689   Median :61.00  
##  Mean   :2021-09-14   Mean   :0.130198   Mean   :59.61  
##  3rd Qu.:2022-01-18   3rd Qu.:0.092545   3rd Qu.:72.00  
##  Max.   :2022-05-25   Max.   :2.855180   Max.   :86.00
colnames(wake_weather)[1]<-"Date"

#mecklenburg weather: data preprocessing#

meck_weather <- read.csv("mecklenburg_weather.csv")
glimpse(meck_weather)
## Rows: 6,940
## Columns: 7
## $ STATION <chr> "US1NCMK0053", "US1NCMK0053", "US1NCMK0053", "US1NCMK0053", "U…
## $ NAME    <chr> "CHARLOTTE 7.0 ENE, NC US", "CHARLOTTE 7.0 ENE, NC US", "CHARL…
## $ DATE    <chr> "2021-01-04", "2021-01-05", "2021-01-06", "2021-01-07", "2021-…
## $ PRCP    <dbl> 0.01, 0.00, 0.04, 0.00, 0.81, 0.00, 0.00, 0.00, 0.00, 0.00, 0.…
## $ TAVG    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TMAX    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TMIN    <int> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
sum(is.na(meck_weather$PRCP))
## [1] 86
meck_weather <- 
  meck_weather %>% mutate(PRCP = replace_na(PRCP,mean(PRCP,na.rm=TRUE)))
          
meck_weather_prcp <- meck_weather %>% group_by(DATE) %>% summarise(mean_precipation = mean(PRCP))
meck_weather_prcp
## # A tibble: 507 × 2
##    DATE       mean_precipation
##    <chr>                 <dbl>
##  1 2021-01-04          0.00231
##  2 2021-01-05          0.0123 
##  3 2021-01-06          0.0192 
##  4 2021-01-07          0.0265 
##  5 2021-01-08          0.505  
##  6 2021-01-09          0.28   
##  7 2021-01-10          0      
##  8 2021-01-11          0.00786
##  9 2021-01-12          0.147  
## 10 2021-01-13          0.0315 
## # … with 497 more rows
sum(is.na(meck_weather$TAVG))
## [1] 6433
full_tavg_data_meck <- meck_weather[complete.cases(meck_weather$TAVG),]
table(full_tavg_data_meck$STATION)
## 
## USW00013881 
##         507
meck_temp <- full_tavg_data_meck %>% select(DATE,TAVG)

meck_weather <- merge(meck_weather_prcp,meck_temp, order.by=DATE)
meck_weather$DATE <- as.Date(meck_weather$DATE)
summary(meck_weather)
##       DATE            mean_precipation       TAVG      
##  Min.   :2021-01-04   Min.   :0.00000   Min.   :28.00  
##  1st Qu.:2021-05-10   1st Qu.:0.00000   1st Qu.:48.00  
##  Median :2021-09-14   Median :0.00500   Median :62.00  
##  Mean   :2021-09-14   Mean   :0.11351   Mean   :60.84  
##  3rd Qu.:2022-01-18   3rd Qu.:0.09521   3rd Qu.:73.00  
##  Max.   :2022-05-25   Max.   :1.77937   Max.   :84.00
colnames(meck_weather)[1]<-"Date"
summary(meck_weather)
##       Date            mean_precipation       TAVG      
##  Min.   :2021-01-04   Min.   :0.00000   Min.   :28.00  
##  1st Qu.:2021-05-10   1st Qu.:0.00000   1st Qu.:48.00  
##  Median :2021-09-14   Median :0.00500   Median :62.00  
##  Mean   :2021-09-14   Mean   :0.11351   Mean   :60.84  
##  3rd Qu.:2022-01-18   3rd Qu.:0.09521   3rd Qu.:73.00  
##  Max.   :2022-05-25   Max.   :1.77937   Max.   :84.00
#new hanover weather: data preprocessing#

hanover_weather<- read.csv("new_hanover_weather.csv")
glimpse(hanover_weather)
## Rows: 10,646
## Columns: 8
## $ STATION <chr> "USC00319461", "USC00319461", "USC00319461", "USC00319461", "U…
## $ NAME    <chr> "WILMINGTON 7 SE, NC US", "WILMINGTON 7 SE, NC US", "WILMINGTO…
## $ DATE    <chr> "2021-01-04", "2021-01-05", "2021-01-06", "2021-01-07", "2021-…
## $ PRCP    <dbl> 0.00, 0.00, 0.04, 0.00, 0.66, 0.06, 0.00, 0.00, 0.26, 0.02, 0.…
## $ TAVG    <lgl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…
## $ TMAX    <int> 68, 52, 52, 51, 51, 54, 50, 51, 55, 49, 55, 57, 61, 52, 52, 57…
## $ TMIN    <int> 47, 36, 36, 32, 32, 37, 31, 31, 33, 38, 38, 36, 36, 32, 32, 31…
## $ TOBS    <int> 47, 39, 36, 32, 43, 37, 32, 33, 45, 38, 42, 36, 41, 32, 42, 31…
sum(is.na(hanover_weather$PRCP))
## [1] 181
hanover_weather <- 
  hanover_weather%>% mutate(PRCP = replace_na(PRCP,mean(PRCP,na.rm=TRUE)))

hanover_weather_prcp <- hanover_weather %>% group_by(DATE) %>% summarise(mean_precipation = mean(PRCP))
hanover_weather_prcp
## # A tibble: 507 × 2
##    DATE       mean_precipation
##    <chr>                 <dbl>
##  1 2021-01-04           0.0123
##  2 2021-01-05           0.0171
##  3 2021-01-06           0.084 
##  4 2021-01-07           0     
##  5 2021-01-08           0.708 
##  6 2021-01-09           0.0533
##  7 2021-01-10           0.0106
##  8 2021-01-11           0.0393
##  9 2021-01-12           0.475 
## 10 2021-01-13           0.0813
## # … with 497 more rows
sum(is.na(hanover_weather$TAVG)) #no data
## [1] 10646
sum(is.na(hanover_weather$TMIN))
## [1] 9222
#which(is.na(hanover_weather$TMIN))
sum(is.na(hanover_weather$TMAX))
## [1] 9221
#which(is.na(hanover_weather$TMAX))

full_temp_data_new_hanover <- 
  hanover_weather[complete.cases(hanover_weather$TMIN),]
full_temp_data_new_hanover 
##          STATION                                    NAME       DATE PRCP TAVG
## 1    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-04 0.00   NA
## 2    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-05 0.00   NA
## 3    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-06 0.04   NA
## 4    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-07 0.00   NA
## 5    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-08 0.66   NA
## 6    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-09 0.06   NA
## 7    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-10 0.00   NA
## 8    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-11 0.00   NA
## 9    USC00319461                  WILMINGTON 7 SE, NC US 2021-01-12 0.26   NA
## 10   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-13 0.02   NA
## 11   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-14 0.20   NA
## 12   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-15 0.00   NA
## 13   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-16 0.26   NA
## 14   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-17 0.00   NA
## 15   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-18 0.00   NA
## 16   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-19 0.00   NA
## 17   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-20 0.00   NA
## 18   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-21 0.00   NA
## 19   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-22 0.00   NA
## 20   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-23 0.00   NA
## 21   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-24 0.00   NA
## 22   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-25 0.02   NA
## 23   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-26 0.00   NA
## 24   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-27 0.18   NA
## 25   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-28 0.30   NA
## 26   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-29 0.00   NA
## 27   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-30 0.00   NA
## 28   USC00319461                  WILMINGTON 7 SE, NC US 2021-01-31 0.00   NA
## 29   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-01 1.62   NA
## 30   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-02 1.60   NA
## 31   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-03 0.00   NA
## 32   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-04 0.00   NA
## 33   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-05 0.00   NA
## 34   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-06 0.00   NA
## 35   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-07 1.32   NA
## 36   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-08 0.00   NA
## 37   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-09 0.00   NA
## 38   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-10 0.06   NA
## 39   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-11 0.10   NA
## 40   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-12 0.15   NA
## 41   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-13 1.10   NA
## 42   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-14 1.28   NA
## 43   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-15 1.10   NA
## 44   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-16 0.30   NA
## 45   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-17 0.00   NA
## 46   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-18 0.02   NA
## 47   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-19 1.12   NA
## 48   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-20 0.50   NA
## 49   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-21 0.00   NA
## 50   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-22 0.06   NA
## 51   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-23 0.00   NA
## 52   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-24 0.00   NA
## 53   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-25 0.00   NA
## 54   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-26 0.00   NA
## 55   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-27 0.00   NA
## 56   USC00319461                  WILMINGTON 7 SE, NC US 2021-02-28 0.00   NA
## 57   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-01 0.00   NA
## 58   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-02 0.00   NA
## 59   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-03 0.20   NA
## 60   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-04 0.41   NA
## 61   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-05 0.00   NA
## 62   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-06 0.00   NA
## 63   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-07 0.00   NA
## 64   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-08 0.00   NA
## 65   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-09 0.00   NA
## 66   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-10 0.00   NA
## 67   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-11 0.00   NA
## 68   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-12 0.00   NA
## 69   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-13 0.00   NA
## 70   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-14 0.00   NA
## 71   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-15 0.04   NA
## 72   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-16 0.02   NA
## 73   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-17 0.90   NA
## 74   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-18 0.00   NA
## 75   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-19 0.00   NA
## 76   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-20 0.00   NA
## 77   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-21 0.00   NA
## 78   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-22 0.00   NA
## 79   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-23 0.00   NA
## 80   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-24 0.10   NA
## 81   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-25 0.00   NA
## 82   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-26 0.00   NA
## 83   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-27 0.00   NA
## 84   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-28 0.00   NA
## 85   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-29 0.08   NA
## 86   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-30 0.00   NA
## 87   USC00319461                  WILMINGTON 7 SE, NC US 2021-03-31 0.00   NA
## 88   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-01 0.80   NA
## 89   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-02 0.00   NA
## 90   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-03 0.00   NA
## 91   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-04 0.00   NA
## 92   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-05 0.00   NA
## 93   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-06 0.00   NA
## 94   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-07 0.00   NA
## 95   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-08 0.00   NA
## 96   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-09 0.00   NA
## 97   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-10 0.00   NA
## 98   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-11 0.44   NA
## 99   USC00319461                  WILMINGTON 7 SE, NC US 2021-04-12 0.00   NA
## 100  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-13 0.00   NA
## 101  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-14 0.00   NA
## 102  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-15 0.00   NA
## 103  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-16 0.01   NA
## 104  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-17 0.00   NA
## 105  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-18 0.00   NA
## 106  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-19 0.00   NA
## 107  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-20 0.00   NA
## 108  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-21 0.00   NA
## 109  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-22 0.00   NA
## 110  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-23 0.00   NA
## 111  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-24 0.00   NA
## 112  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-25 0.06   NA
## 113  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-26 0.00   NA
## 114  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-27 0.00   NA
## 115  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-28 0.00   NA
## 116  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-29 0.00   NA
## 117  USC00319461                  WILMINGTON 7 SE, NC US 2021-04-30 0.00   NA
## 118  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-01 0.00   NA
## 119  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-02 0.00   NA
## 120  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-03 0.00   NA
## 121  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-04 0.00   NA
## 122  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-05 0.00   NA
## 123  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-06 0.00   NA
## 124  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-07 0.00   NA
## 125  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-08 0.74   NA
## 126  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-09 0.00   NA
## 127  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-10 0.00   NA
## 128  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-11 0.02   NA
## 129  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-12 0.04   NA
## 130  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-13 0.28   NA
## 131  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-14 0.00   NA
## 132  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-15 0.00   NA
## 133  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-16 0.00   NA
## 134  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-17 0.00   NA
## 135  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-18 0.00   NA
## 136  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-19 0.00   NA
## 137  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-20 0.00   NA
## 138  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-21 0.00   NA
## 139  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-22 0.00   NA
## 140  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-23 0.00   NA
## 141  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-24 0.00   NA
## 142  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-25 0.22   NA
## 143  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-26 0.00   NA
## 144  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-28 0.00   NA
## 145  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-29 0.00   NA
## 146  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-30 0.38   NA
## 147  USC00319461                  WILMINGTON 7 SE, NC US 2021-05-31 0.00   NA
## 148  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-01 0.00   NA
## 149  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-02 0.00   NA
## 150  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-03 0.22   NA
## 151  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-04 3.26   NA
## 152  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-05 0.12   NA
## 153  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-06 0.00   NA
## 154  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-07 0.34   NA
## 155  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-08 0.00   NA
## 156  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-09 0.00   NA
## 157  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-10 0.08   NA
## 158  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-11 0.12   NA
## 159  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-12 0.16   NA
## 160  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-13 1.08   NA
## 161  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-14 0.02   NA
## 162  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-15 0.00   NA
## 163  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-16 2.12   NA
## 164  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-17 0.00   NA
## 165  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-18 0.00   NA
## 166  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-19 0.00   NA
## 167  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-20 0.62   NA
## 168  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-21 0.72   NA
## 169  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-22 0.00   NA
## 170  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-23 0.54   NA
## 171  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-24 0.00   NA
## 172  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-25 0.94   NA
## 173  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-26 0.24   NA
## 174  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-27 0.00   NA
## 175  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-28 0.00   NA
## 176  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-29 0.00   NA
## 177  USC00319461                  WILMINGTON 7 SE, NC US 2021-06-30 0.00   NA
## 178  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-01 0.00   NA
## 179  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-02 0.00   NA
## 180  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-03 0.88   NA
## 181  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-04 0.00   NA
## 182  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-05 0.00   NA
## 183  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-06 0.00   NA
## 184  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-07 0.00   NA
## 185  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-08 0.04   NA
## 186  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-09 1.05   NA
## 187  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-10 0.00   NA
## 188  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-11 0.00   NA
## 189  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-12 0.00   NA
## 190  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-13 0.00   NA
## 191  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-14 0.00   NA
## 192  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-15 0.00   NA
## 193  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-16 0.06   NA
## 194  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-17 0.00   NA
## 195  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-18 0.00   NA
## 196  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-19 1.08   NA
## 197  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-20 2.52   NA
## 198  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-21 0.00   NA
## 199  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-22 0.00   NA
## 200  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-23 0.00   NA
## 201  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-24 0.00   NA
## 202  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-25 0.00   NA
## 203  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-26 0.00   NA
## 204  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-27 0.04   NA
## 205  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-28 0.58   NA
## 206  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-29 0.00   NA
## 207  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-30 0.00   NA
## 208  USC00319461                  WILMINGTON 7 SE, NC US 2021-07-31 0.00   NA
## 209  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-01 0.00   NA
## 210  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-02 0.26   NA
## 211  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-03 0.82   NA
## 212  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-04 3.42   NA
## 213  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-05 0.00   NA
## 214  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-06 0.00   NA
## 215  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-07 0.62   NA
## 216  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-08 4.10   NA
## 217  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-09 0.00   NA
## 218  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-10 0.00   NA
## 219  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-11 0.00   NA
## 220  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-12 0.00   NA
## 221  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-13 0.00   NA
## 222  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-14 0.00   NA
## 223  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-15 0.00   NA
## 224  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-16 1.24   NA
## 225  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-17 0.08   NA
## 226  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-18 1.62   NA
## 227  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-19 0.00   NA
## 228  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-20 0.00   NA
## 229  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-21 2.76   NA
## 230  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-22 0.16   NA
## 231  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-23 0.16   NA
## 232  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-24 0.00   NA
## 233  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-25 0.14   NA
## 234  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-26 0.00   NA
## 235  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-27 0.00   NA
## 236  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-28 0.00   NA
## 237  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-29 0.00   NA
## 239  USC00319461                  WILMINGTON 7 SE, NC US 2021-08-31 0.00   NA
## 240  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-01 0.04   NA
## 241  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-02 0.16   NA
## 242  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-03 0.00   NA
## 243  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-04 0.00   NA
## 244  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-05 0.00   NA
## 245  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-06 0.00   NA
## 246  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-07 0.04   NA
## 247  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-08 0.32   NA
## 248  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-09 0.16   NA
## 249  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-10 0.52   NA
## 250  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-11 0.00   NA
## 251  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-12 0.00   NA
## 252  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-13 0.00   NA
## 253  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-14 0.00   NA
## 254  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-15 0.00   NA
## 255  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-16 0.00   NA
## 256  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-17 0.12   NA
## 257  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-18 0.00   NA
## 258  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-19 0.00   NA
## 259  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-20 0.00   NA
## 260  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-21 0.72   NA
## 261  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-22 5.16   NA
## 262  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-23 3.02   NA
## 263  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-24 0.00   NA
## 264  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-25 0.00   NA
## 265  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-26 0.00   NA
## 266  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-27 0.00   NA
## 267  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-28 0.00   NA
## 268  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-29 0.00   NA
## 269  USC00319461                  WILMINGTON 7 SE, NC US 2021-09-30 0.00   NA
## 270  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-01 0.00   NA
## 271  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-02 0.00   NA
## 272  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-03 0.00   NA
## 273  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-04 0.16   NA
## 274  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-05 0.00   NA
## 275  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-06 0.04   NA
## 276  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-07 0.14   NA
## 277  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-08 0.98   NA
## 278  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-09 0.00   NA
## 279  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-10 0.06   NA
## 280  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-11 0.00   NA
## 281  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-12 0.00   NA
## 282  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-13 0.00   NA
## 283  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-14 0.00   NA
## 284  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-15 0.00   NA
## 285  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-16 0.00   NA
## 286  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-17 0.00   NA
## 287  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-18 0.00   NA
## 288  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-19 0.00   NA
## 289  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-20 0.00   NA
## 290  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-21 0.00   NA
## 291  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-22 0.00   NA
## 292  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-23 0.00   NA
## 293  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-24 0.00   NA
## 294  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-25 0.00   NA
## 295  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-26 0.12   NA
## 296  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-27 0.00   NA
## 297  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-28 0.00   NA
## 298  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-29 0.20   NA
## 299  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-30 0.04   NA
## 300  USC00319461                  WILMINGTON 7 SE, NC US 2021-10-31 0.00   NA
## 301  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-01 0.00   NA
## 302  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-02 0.00   NA
## 303  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-03 0.00   NA
## 304  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-04 0.00   NA
## 305  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-05 0.00   NA
## 306  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-06 0.00   NA
## 307  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-07 0.78   NA
## 308  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-08 0.00   NA
## 309  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-09 0.00   NA
## 310  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-10 0.00   NA
## 311  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-11 0.00   NA
## 312  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-12 0.18   NA
## 313  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-13 0.00   NA
## 314  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-14 0.00   NA
## 315  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-15 0.00   NA
## 316  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-16 0.00   NA
## 317  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-17 0.00   NA
## 318  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-18 0.00   NA
## 319  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-19 0.00   NA
## 320  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-20 0.00   NA
## 321  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-21 0.00   NA
## 322  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-22 0.00   NA
## 323  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-23 0.01   NA
## 324  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-24 0.00   NA
## 325  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-25 0.00   NA
## 326  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-26 0.02   NA
## 327  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-27 0.00   NA
## 328  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-28 0.00   NA
## 329  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-29 0.00   NA
## 330  USC00319461                  WILMINGTON 7 SE, NC US 2021-11-30 0.00   NA
## 331  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-01 0.00   NA
## 332  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-02 0.00   NA
## 333  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-03 0.00   NA
## 334  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-04 0.00   NA
## 335  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-05 0.00   NA
## 336  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-06 0.00   NA
## 337  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-07 0.00   NA
## 338  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-08 0.00   NA
## 339  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-09 1.40   NA
## 340  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-10 0.00   NA
## 341  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-11 0.04   NA
## 342  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-12 0.24   NA
## 343  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-13 0.00   NA
## 344  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-14 0.00   NA
## 345  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-15 0.00   NA
## 346  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-16 0.00   NA
## 347  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-17 0.00   NA
## 348  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-18 0.00   NA
## 349  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-19 0.00   NA
## 350  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-20 0.40   NA
## 351  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-21 0.00   NA
## 352  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-22 1.18   NA
## 353  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-23 0.02   NA
## 354  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-24 0.00   NA
## 355  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-25 0.00   NA
## 356  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-26 0.00   NA
## 357  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-27 0.00   NA
## 358  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-28 0.00   NA
## 359  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-29 0.00   NA
## 360  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-30 0.00   NA
## 361  USC00319461                  WILMINGTON 7 SE, NC US 2021-12-31 0.00   NA
## 362  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-01 0.00   NA
## 363  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-02 0.00   NA
## 364  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-03 0.66   NA
## 365  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-04 0.00   NA
## 366  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-05 0.00   NA
## 367  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-06 0.12   NA
## 368  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-07 0.00   NA
## 369  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-08 0.00   NA
## 370  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-09 0.00   NA
## 371  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-10 0.12   NA
## 372  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-11 0.00   NA
## 373  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-12 0.00   NA
## 374  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-13 0.00   NA
## 375  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-14 0.00   NA
## 376  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-15 0.00   NA
## 377  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-16 0.16   NA
## 378  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-17 1.64   NA
## 379  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-18 0.00   NA
## 380  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-19 0.00   NA
## 381  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-20 0.00   NA
## 382  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-21 0.22   NA
## 383  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-22 0.26   NA
## 384  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-23 0.00   NA
## 385  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-24 0.00   NA
## 386  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-25 0.00   NA
## 387  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-26 0.00   NA
## 388  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-27 0.00   NA
## 389  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-28 0.00   NA
## 390  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-29 0.06   NA
## 391  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-30 0.04   NA
## 392  USC00319461                  WILMINGTON 7 SE, NC US 2022-01-31 0.00   NA
## 393  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-01 0.00   NA
## 394  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-02 0.00   NA
## 395  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-03 0.02   NA
## 396  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-04 0.04   NA
## 397  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-05 0.36   NA
## 398  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-06 0.00   NA
## 399  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-07 0.00   NA
## 400  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-08 0.22   NA
## 401  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-09 0.00   NA
## 402  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-10 0.00   NA
## 403  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-11 0.00   NA
## 404  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-12 0.00   NA
## 405  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-13 0.00   NA
## 406  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-14 0.12   NA
## 407  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-15 0.00   NA
## 408  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-16 0.00   NA
## 409  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-17 0.00   NA
## 410  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-18 0.16   NA
## 411  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-19 0.24   NA
## 412  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-20 0.00   NA
## 413  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-21 0.00   NA
## 414  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-22 0.00   NA
## 415  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-23 0.00   NA
## 416  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-24 0.00   NA
## 417  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-25 0.00   NA
## 418  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-26 0.00   NA
## 419  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-27 0.00   NA
## 420  USC00319461                  WILMINGTON 7 SE, NC US 2022-02-28 0.31   NA
## 421  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-01 0.00   NA
## 422  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-02 0.00   NA
## 423  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-03 0.00   NA
## 424  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-04 0.00   NA
## 425  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-05 0.00   NA
## 426  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-06 0.00   NA
## 427  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-07 0.00   NA
## 428  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-08 0.00   NA
## 429  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-09 0.16   NA
## 430  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-10 0.18   NA
## 431  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-11 0.00   NA
## 432  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-12 0.18   NA
## 433  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-13 0.26   NA
## 434  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-14 0.00   NA
## 435  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-15 0.00   NA
## 436  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-16 0.00   NA
## 437  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-17 0.10   NA
## 438  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-18 0.00   NA
## 439  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-19 0.00   NA
## 440  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-20 0.00   NA
## 441  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-21 0.00   NA
## 442  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-22 0.00   NA
## 443  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-23 0.00   NA
## 444  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-24 0.40   NA
## 445  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-25 2.52   NA
## 446  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-26 0.00   NA
## 447  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-27 0.00   NA
## 448  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-28 0.00   NA
## 449  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-29 0.00   NA
## 450  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-30 0.00   NA
## 451  USC00319461                  WILMINGTON 7 SE, NC US 2022-03-31 0.00   NA
## 452  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-01 0.24   NA
## 453  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-02 0.00   NA
## 454  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-03 0.00   NA
## 455  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-04 0.00   NA
## 456  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-05 0.00   NA
## 457  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-06 0.62   NA
## 458  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-07 0.32   NA
## 459  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-08 0.00   NA
## 460  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-09 0.00   NA
## 461  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-10 0.00   NA
## 462  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-11 0.00   NA
## 463  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-12 0.00   NA
## 464  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-13 0.00   NA
## 465  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-14 0.00   NA
## 466  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-15 0.00   NA
## 467  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-16 0.00   NA
## 468  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-17 0.22   NA
## 469  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-18 1.24   NA
## 470  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-19 0.00   NA
## 471  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-20 0.00   NA
## 472  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-21 0.00   NA
## 473  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-22 0.00   NA
## 474  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-23 0.00   NA
## 475  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-24 0.00   NA
## 476  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-25 0.00   NA
## 477  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-26 0.00   NA
## 478  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-27 0.04   NA
## 479  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-28 0.00   NA
## 480  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-29 0.00   NA
## 481  USC00319461                  WILMINGTON 7 SE, NC US 2022-04-30 0.00   NA
## 482  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-01 0.00   NA
## 483  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-02 0.00   NA
## 484  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-03 0.00   NA
## 485  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-04 0.00   NA
## 486  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-05 0.08   NA
## 487  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-06 0.00   NA
## 488  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-07 0.00   NA
## 489  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-08 0.00   NA
## 490  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-09 0.00   NA
## 491  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-10 0.00   NA
## 492  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-11 0.00   NA
## 493  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-12 0.03   NA
## 494  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-13 0.22   NA
## 495  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-14 0.14   NA
## 496  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-15 0.12   NA
## 497  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-16 0.00   NA
## 498  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-17 0.32   NA
## 499  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-18 0.00   NA
## 500  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-19 0.00   NA
## 501  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-20 0.00   NA
## 502  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-21 0.00   NA
## 503  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-22 0.00   NA
## 504  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-23 0.08   NA
## 505  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-24 0.00   NA
## 506  USC00319461                  WILMINGTON 7 SE, NC US 2022-05-25 0.00   NA
## 2059 USC00319467                   WILMINGTON 7 N, NC US 2021-01-04 0.00   NA
## 2060 USC00319467                   WILMINGTON 7 N, NC US 2021-01-05 0.10   NA
## 2061 USC00319467                   WILMINGTON 7 N, NC US 2021-01-06 0.00   NA
## 2062 USC00319467                   WILMINGTON 7 N, NC US 2021-01-07 0.00   NA
## 2063 USC00319467                   WILMINGTON 7 N, NC US 2021-01-08 0.73   NA
## 2064 USC00319467                   WILMINGTON 7 N, NC US 2021-01-09 0.00   NA
## 2065 USC00319467                   WILMINGTON 7 N, NC US 2021-01-10 0.00   NA
## 2066 USC00319467                   WILMINGTON 7 N, NC US 2021-01-11 0.20   NA
## 2067 USC00319467                   WILMINGTON 7 N, NC US 2021-01-12 0.32   NA
## 2068 USC00319467                   WILMINGTON 7 N, NC US 2021-01-13 0.00   NA
## 2069 USC00319467                   WILMINGTON 7 N, NC US 2021-01-14 0.09   NA
## 2070 USC00319467                   WILMINGTON 7 N, NC US 2021-01-15 0.26   NA
## 2071 USC00319467                   WILMINGTON 7 N, NC US 2021-01-16 0.01   NA
## 2072 USC00319467                   WILMINGTON 7 N, NC US 2021-01-17 0.00   NA
## 2073 USC00319467                   WILMINGTON 7 N, NC US 2021-01-18 0.00   NA
## 2074 USC00319467                   WILMINGTON 7 N, NC US 2021-01-19 0.00   NA
## 2075 USC00319467                   WILMINGTON 7 N, NC US 2021-01-20 0.00   NA
## 2076 USC00319467                   WILMINGTON 7 N, NC US 2021-01-21 0.00   NA
## 2077 USC00319467                   WILMINGTON 7 N, NC US 2021-01-22 0.00   NA
## 2078 USC00319467                   WILMINGTON 7 N, NC US 2021-01-23 0.00   NA
## 2079 USC00319467                   WILMINGTON 7 N, NC US 2021-01-24 0.00   NA
## 2080 USC00319467                   WILMINGTON 7 N, NC US 2021-01-25 0.57   NA
## 2081 USC00319467                   WILMINGTON 7 N, NC US 2021-01-26 0.11   NA
## 2082 USC00319467                   WILMINGTON 7 N, NC US 2021-01-27 0.37   NA
## 2083 USC00319467                   WILMINGTON 7 N, NC US 2021-01-28 0.19   NA
## 2084 USC00319467                   WILMINGTON 7 N, NC US 2021-01-29 0.00   NA
## 2085 USC00319467                   WILMINGTON 7 N, NC US 2021-01-30 0.00   NA
## 2086 USC00319467                   WILMINGTON 7 N, NC US 2021-01-31 1.42   NA
## 2087 USC00319467                   WILMINGTON 7 N, NC US 2021-02-01 0.00   NA
## 2088 USC00319467                   WILMINGTON 7 N, NC US 2021-02-02 0.00   NA
## 2089 USC00319467                   WILMINGTON 7 N, NC US 2021-02-03 0.00   NA
## 2090 USC00319467                   WILMINGTON 7 N, NC US 2021-02-04 0.00   NA
## 2091 USC00319467                   WILMINGTON 7 N, NC US 2021-02-05 0.15   NA
## 2092 USC00319467                   WILMINGTON 7 N, NC US 2021-02-06 0.09   NA
## 2093 USC00319467                   WILMINGTON 7 N, NC US 2021-02-07 0.65   NA
## 2094 USC00319467                   WILMINGTON 7 N, NC US 2021-02-08 0.00   NA
## 2095 USC00319467                   WILMINGTON 7 N, NC US 2021-02-09 0.14   NA
## 2096 USC00319467                   WILMINGTON 7 N, NC US 2021-02-10 0.11   NA
## 2097 USC00319467                   WILMINGTON 7 N, NC US 2021-02-11 0.08   NA
## 2098 USC00319467                   WILMINGTON 7 N, NC US 2021-02-12 0.25   NA
## 2099 USC00319467                   WILMINGTON 7 N, NC US 2021-02-13 0.06   NA
## 2100 USC00319467                   WILMINGTON 7 N, NC US 2021-02-14 0.18   NA
## 2101 USC00319467                   WILMINGTON 7 N, NC US 2021-02-15 0.00   NA
## 2102 USC00319467                   WILMINGTON 7 N, NC US 2021-02-16 0.00   NA
## 2103 USC00319467                   WILMINGTON 7 N, NC US 2021-02-17 0.00   NA
## 2104 USC00319467                   WILMINGTON 7 N, NC US 2021-02-18 0.00   NA
## 2105 USC00319467                   WILMINGTON 7 N, NC US 2021-02-19 0.31   NA
## 2106 USC00319467                   WILMINGTON 7 N, NC US 2021-02-20 0.00   NA
## 2107 USC00319467                   WILMINGTON 7 N, NC US 2021-02-21 0.00   NA
## 2108 USC00319467                   WILMINGTON 7 N, NC US 2021-02-22 0.17   NA
## 2109 USC00319467                   WILMINGTON 7 N, NC US 2021-02-23 0.00   NA
## 2110 USC00319467                   WILMINGTON 7 N, NC US 2021-02-24 0.00   NA
## 2111 USC00319467                   WILMINGTON 7 N, NC US 2021-02-25 0.00   NA
## 2112 USC00319467                   WILMINGTON 7 N, NC US 2021-02-26 0.00   NA
## 2113 USC00319467                   WILMINGTON 7 N, NC US 2021-02-27 0.04   NA
## 2114 USC00319467                   WILMINGTON 7 N, NC US 2021-02-28 0.00   NA
## 2115 USC00319467                   WILMINGTON 7 N, NC US 2021-03-03 0.01   NA
## 2116 USC00319467                   WILMINGTON 7 N, NC US 2021-03-04 0.00   NA
## 2117 USC00319467                   WILMINGTON 7 N, NC US 2021-03-05 0.00   NA
## 2118 USC00319467                   WILMINGTON 7 N, NC US 2021-03-06 0.00   NA
## 2119 USC00319467                   WILMINGTON 7 N, NC US 2021-03-07 0.00   NA
## 2120 USC00319467                   WILMINGTON 7 N, NC US 2021-03-08 0.00   NA
## 2121 USC00319467                   WILMINGTON 7 N, NC US 2021-03-09 0.00   NA
## 2122 USC00319467                   WILMINGTON 7 N, NC US 2021-03-10 0.00   NA
## 2123 USC00319467                   WILMINGTON 7 N, NC US 2021-03-11 0.00   NA
## 2124 USC00319467                   WILMINGTON 7 N, NC US 2021-03-12 0.00   NA
## 2125 USC00319467                   WILMINGTON 7 N, NC US 2021-03-13 0.00   NA
## 2126 USC00319467                   WILMINGTON 7 N, NC US 2021-03-14 0.01   NA
## 2127 USC00319467                   WILMINGTON 7 N, NC US 2021-03-15 0.00   NA
## 2128 USC00319467                   WILMINGTON 7 N, NC US 2021-03-16 0.28   NA
## 2129 USC00319467                   WILMINGTON 7 N, NC US 2021-03-17 0.52   NA
## 2130 USC00319467                   WILMINGTON 7 N, NC US 2021-03-18 0.00   NA
## 2131 USC00319467                   WILMINGTON 7 N, NC US 2021-03-19 0.00   NA
## 2132 USC00319467                   WILMINGTON 7 N, NC US 2021-03-20 0.07   NA
## 2133 USC00319467                   WILMINGTON 7 N, NC US 2021-03-21 0.00   NA
## 2134 USC00319467                   WILMINGTON 7 N, NC US 2021-03-22 0.00   NA
## 2135 USC00319467                   WILMINGTON 7 N, NC US 2021-03-23 0.01   NA
## 2136 USC00319467                   WILMINGTON 7 N, NC US 2021-03-24 0.09   NA
## 2137 USC00319467                   WILMINGTON 7 N, NC US 2021-03-25 0.00   NA
## 2138 USC00319467                   WILMINGTON 7 N, NC US 2021-03-26 0.00   NA
## 2139 USC00319467                   WILMINGTON 7 N, NC US 2021-03-27 0.00   NA
## 2140 USC00319467                   WILMINGTON 7 N, NC US 2021-03-28 0.17   NA
## 2141 USC00319467                   WILMINGTON 7 N, NC US 2021-03-29 0.00   NA
## 2142 USC00319467                   WILMINGTON 7 N, NC US 2021-03-30 0.00   NA
## 2143 USC00319467                   WILMINGTON 7 N, NC US 2021-04-01 0.00   NA
## 2144 USC00319467                   WILMINGTON 7 N, NC US 2021-04-02 0.00   NA
## 2145 USC00319467                   WILMINGTON 7 N, NC US 2021-04-03 0.00   NA
## 2146 USC00319467                   WILMINGTON 7 N, NC US 2021-04-04 0.00   NA
## 2147 USC00319467                   WILMINGTON 7 N, NC US 2021-04-05 0.00   NA
## 2148 USC00319467                   WILMINGTON 7 N, NC US 2021-04-06 0.00   NA
## 2149 USC00319467                   WILMINGTON 7 N, NC US 2021-04-07 0.00   NA
## 2150 USC00319467                   WILMINGTON 7 N, NC US 2021-04-08 0.00   NA
## 2151 USC00319467                   WILMINGTON 7 N, NC US 2021-04-09 0.00   NA
## 2152 USC00319467                   WILMINGTON 7 N, NC US 2021-04-10 0.00   NA
## 2153 USC00319467                   WILMINGTON 7 N, NC US 2021-04-11 0.49   NA
## 2154 USC00319467                   WILMINGTON 7 N, NC US 2021-04-12 0.00   NA
## 2155 USC00319467                   WILMINGTON 7 N, NC US 2021-04-13 0.00   NA
## 2156 USC00319467                   WILMINGTON 7 N, NC US 2021-04-14 0.00   NA
## 2157 USC00319467                   WILMINGTON 7 N, NC US 2021-04-15 0.10   NA
## 2158 USC00319467                   WILMINGTON 7 N, NC US 2021-04-16 0.00   NA
## 2159 USC00319467                   WILMINGTON 7 N, NC US 2021-04-17 0.00   NA
## 2160 USC00319467                   WILMINGTON 7 N, NC US 2021-04-18 0.00   NA
## 2161 USC00319467                   WILMINGTON 7 N, NC US 2021-04-19 0.00   NA
## 2162 USC00319467                   WILMINGTON 7 N, NC US 2021-04-20 0.00   NA
## 2163 USC00319467                   WILMINGTON 7 N, NC US 2021-04-21 0.00   NA
## 2164 USC00319467                   WILMINGTON 7 N, NC US 2021-04-22 0.00   NA
## 2165 USC00319467                   WILMINGTON 7 N, NC US 2021-04-23 0.00   NA
## 2166 USC00319467                   WILMINGTON 7 N, NC US 2021-04-24 0.00   NA
## 2167 USC00319467                   WILMINGTON 7 N, NC US 2021-04-25 0.34   NA
## 2168 USC00319467                   WILMINGTON 7 N, NC US 2021-04-26 0.00   NA
## 2169 USC00319467                   WILMINGTON 7 N, NC US 2021-04-27 0.00   NA
## 2170 USC00319467                   WILMINGTON 7 N, NC US 2021-04-28 0.00   NA
## 2171 USC00319467                   WILMINGTON 7 N, NC US 2021-04-29 0.00   NA
## 2172 USC00319467                   WILMINGTON 7 N, NC US 2021-04-30 0.00   NA
## 2173 USC00319467                   WILMINGTON 7 N, NC US 2021-05-01 0.00   NA
## 2174 USC00319467                   WILMINGTON 7 N, NC US 2021-05-02 0.00   NA
## 2175 USC00319467                   WILMINGTON 7 N, NC US 2021-05-03 0.06   NA
## 2176 USC00319467                   WILMINGTON 7 N, NC US 2021-05-04 0.00   NA
## 2177 USC00319467                   WILMINGTON 7 N, NC US 2021-05-05 0.00   NA
## 2178 USC00319467                   WILMINGTON 7 N, NC US 2021-05-06 0.00   NA
## 2179 USC00319467                   WILMINGTON 7 N, NC US 2021-05-07 1.04   NA
## 2180 USC00319467                   WILMINGTON 7 N, NC US 2021-05-08 0.00   NA
## 2181 USC00319467                   WILMINGTON 7 N, NC US 2021-05-09 0.00   NA
## 2182 USC00319467                   WILMINGTON 7 N, NC US 2021-05-10 0.04   NA
## 2183 USC00319467                   WILMINGTON 7 N, NC US 2021-05-11 0.00   NA
## 2184 USC00319467                   WILMINGTON 7 N, NC US 2021-05-12 0.52   NA
## 2185 USC00319467                   WILMINGTON 7 N, NC US 2021-05-13 0.00   NA
## 2186 USC00319467                   WILMINGTON 7 N, NC US 2021-05-14 0.00   NA
## 2187 USC00319467                   WILMINGTON 7 N, NC US 2021-05-15 0.00   NA
## 2188 USC00319467                   WILMINGTON 7 N, NC US 2021-05-16 0.00   NA
## 2189 USC00319467                   WILMINGTON 7 N, NC US 2021-05-17 0.00   NA
## 2190 USC00319467                   WILMINGTON 7 N, NC US 2021-05-18 0.00   NA
## 2191 USC00319467                   WILMINGTON 7 N, NC US 2021-05-19 0.00   NA
## 2192 USC00319467                   WILMINGTON 7 N, NC US 2021-05-20 0.00   NA
## 2193 USC00319467                   WILMINGTON 7 N, NC US 2021-05-21 0.00   NA
## 2194 USC00319467                   WILMINGTON 7 N, NC US 2021-05-22 0.00   NA
## 2195 USC00319467                   WILMINGTON 7 N, NC US 2021-05-23 0.00   NA
## 2196 USC00319467                   WILMINGTON 7 N, NC US 2021-05-24 0.08   NA
## 2197 USC00319467                   WILMINGTON 7 N, NC US 2021-05-25 0.05   NA
## 2198 USC00319467                   WILMINGTON 7 N, NC US 2021-05-26 0.00   NA
## 2199 USC00319467                   WILMINGTON 7 N, NC US 2021-05-27 0.00   NA
## 2200 USC00319467                   WILMINGTON 7 N, NC US 2021-05-28 0.27   NA
## 2201 USC00319467                   WILMINGTON 7 N, NC US 2021-05-29 0.00   NA
## 2202 USC00319467                   WILMINGTON 7 N, NC US 2021-05-30 0.18   NA
## 2203 USC00319467                   WILMINGTON 7 N, NC US 2021-05-31 0.00   NA
## 2204 USC00319467                   WILMINGTON 7 N, NC US 2021-06-01 0.00   NA
## 2205 USC00319467                   WILMINGTON 7 N, NC US 2021-06-02 1.07   NA
## 2206 USC00319467                   WILMINGTON 7 N, NC US 2021-06-03 2.00   NA
## 2207 USC00319467                   WILMINGTON 7 N, NC US 2021-07-01 0.00   NA
## 2208 USC00319467                   WILMINGTON 7 N, NC US 2021-07-02 3.27   NA
## 2209 USC00319467                   WILMINGTON 7 N, NC US 2021-07-03 0.00   NA
## 2210 USC00319467                   WILMINGTON 7 N, NC US 2021-07-04 0.00   NA
## 2211 USC00319467                   WILMINGTON 7 N, NC US 2021-07-05 0.00   NA
## 2212 USC00319467                   WILMINGTON 7 N, NC US 2021-07-06 0.00   NA
## 2213 USC00319467                   WILMINGTON 7 N, NC US 2021-07-07 0.10   NA
## 2214 USC00319467                   WILMINGTON 7 N, NC US 2021-07-08 0.83   NA
## 2215 USC00319467                   WILMINGTON 7 N, NC US 2021-07-09 1.18   NA
## 2216 USC00319467                   WILMINGTON 7 N, NC US 2021-07-10 0.00   NA
## 2217 USC00319467                   WILMINGTON 7 N, NC US 2021-07-11 0.19   NA
## 2218 USC00319467                   WILMINGTON 7 N, NC US 2021-07-12 0.04   NA
## 2219 USC00319467                   WILMINGTON 7 N, NC US 2021-07-13 0.03   NA
## 2220 USC00319467                   WILMINGTON 7 N, NC US 2021-07-14 0.02   NA
## 2221 USC00319467                   WILMINGTON 7 N, NC US 2021-07-15 0.00   NA
## 2222 USC00319467                   WILMINGTON 7 N, NC US 2021-07-16 0.03   NA
## 2223 USC00319467                   WILMINGTON 7 N, NC US 2021-07-17 0.00   NA
## 2224 USC00319467                   WILMINGTON 7 N, NC US 2021-07-18 2.04   NA
## 2225 USC00319467                   WILMINGTON 7 N, NC US 2021-07-19 1.56   NA
## 2226 USC00319467                   WILMINGTON 7 N, NC US 2021-07-20 0.02   NA
## 2227 USC00319467                   WILMINGTON 7 N, NC US 2021-07-21 0.02   NA
## 2228 USC00319467                   WILMINGTON 7 N, NC US 2021-07-22 0.20   NA
## 2229 USC00319467                   WILMINGTON 7 N, NC US 2021-07-23 0.00   NA
## 2230 USC00319467                   WILMINGTON 7 N, NC US 2021-07-24 0.00   NA
## 2231 USC00319467                   WILMINGTON 7 N, NC US 2021-07-25 0.00   NA
## 2232 USC00319467                   WILMINGTON 7 N, NC US 2021-07-26 0.07   NA
## 2233 USC00319467                   WILMINGTON 7 N, NC US 2021-07-27 0.00   NA
## 2234 USC00319467                   WILMINGTON 7 N, NC US 2021-07-28 0.21   NA
## 2235 USC00319467                   WILMINGTON 7 N, NC US 2021-07-29 0.00   NA
## 2236 USC00319467                   WILMINGTON 7 N, NC US 2021-07-30 0.00   NA
## 2237 USC00319467                   WILMINGTON 7 N, NC US 2021-07-31 0.61   NA
## 2238 USC00319467                   WILMINGTON 7 N, NC US 2021-08-01 0.53   NA
## 2239 USC00319467                   WILMINGTON 7 N, NC US 2021-08-02 3.35   NA
## 2240 USC00319467                   WILMINGTON 7 N, NC US 2021-08-03 2.38   NA
## 2241 USC00319467                   WILMINGTON 7 N, NC US 2021-08-05 0.00   NA
## 2242 USC00319467                   WILMINGTON 7 N, NC US 2021-08-06 1.75   NA
## 2243 USC00319467                   WILMINGTON 7 N, NC US 2021-08-07 0.83   NA
## 2244 USC00319467                   WILMINGTON 7 N, NC US 2021-08-08 0.00   NA
## 2245 USC00319467                   WILMINGTON 7 N, NC US 2021-08-09 0.00   NA
## 2246 USC00319467                   WILMINGTON 7 N, NC US 2021-08-10 0.00   NA
## 2247 USC00319467                   WILMINGTON 7 N, NC US 2021-08-11 0.00   NA
## 2248 USC00319467                   WILMINGTON 7 N, NC US 2021-08-12 0.00   NA
## 2249 USC00319467                   WILMINGTON 7 N, NC US 2021-08-13 0.00   NA
## 2250 USC00319467                   WILMINGTON 7 N, NC US 2021-08-14 0.00   NA
## 2251 USC00319467                   WILMINGTON 7 N, NC US 2021-08-15 0.00   NA
## 2252 USC00319467                   WILMINGTON 7 N, NC US 2021-08-16 0.09   NA
## 2253 USC00319467                   WILMINGTON 7 N, NC US 2021-08-17 1.59   NA
## 2254 USC00319467                   WILMINGTON 7 N, NC US 2021-08-18 1.03   NA
## 2255 USC00319467                   WILMINGTON 7 N, NC US 2021-08-19 0.00   NA
## 2256 USC00319467                   WILMINGTON 7 N, NC US 2021-08-20 1.37   NA
## 2257 USC00319467                   WILMINGTON 7 N, NC US 2021-08-21 0.47   NA
## 2258 USC00319467                   WILMINGTON 7 N, NC US 2021-09-01 0.11   NA
## 2259 USC00319467                   WILMINGTON 7 N, NC US 2021-09-02 0.02   NA
## 2260 USC00319467                   WILMINGTON 7 N, NC US 2021-09-03 0.00   NA
## 2261 USC00319467                   WILMINGTON 7 N, NC US 2021-09-04 0.00   NA
## 2262 USC00319467                   WILMINGTON 7 N, NC US 2021-09-05 0.00   NA
## 2263 USC00319467                   WILMINGTON 7 N, NC US 2021-09-06 0.01   NA
## 2264 USC00319467                   WILMINGTON 7 N, NC US 2021-09-07 1.50   NA
## 2265 USC00319467                   WILMINGTON 7 N, NC US 2021-09-08 0.04   NA
## 2266 USC00319467                   WILMINGTON 7 N, NC US 2021-09-09 0.52   NA
## 2267 USC00319467                   WILMINGTON 7 N, NC US 2021-09-10 0.00   NA
## 2268 USC00319467                   WILMINGTON 7 N, NC US 2021-09-11 0.01   NA
## 2269 USC00319467                   WILMINGTON 7 N, NC US 2021-09-12 0.00   NA
## 2270 USC00319467                   WILMINGTON 7 N, NC US 2021-09-13 0.00   NA
## 2271 USC00319467                   WILMINGTON 7 N, NC US 2021-09-14 0.00   NA
## 2272 USC00319467                   WILMINGTON 7 N, NC US 2021-09-15 0.00   NA
## 2273 USC00319467                   WILMINGTON 7 N, NC US 2021-09-16 0.00   NA
## 2274 USC00319467                   WILMINGTON 7 N, NC US 2021-09-17 0.00   NA
## 2275 USC00319467                   WILMINGTON 7 N, NC US 2021-09-18 0.00   NA
## 2276 USC00319467                   WILMINGTON 7 N, NC US 2021-09-19 0.00   NA
## 2277 USC00319467                   WILMINGTON 7 N, NC US 2021-09-20 2.52   NA
## 2278 USC00319467                   WILMINGTON 7 N, NC US 2021-09-21 3.39   NA
## 2279 USC00319467                   WILMINGTON 7 N, NC US 2021-09-22 3.42   NA
## 2280 USC00319467                   WILMINGTON 7 N, NC US 2021-09-23 0.36   NA
## 2281 USC00319467                   WILMINGTON 7 N, NC US 2021-09-24 0.00   NA
## 2282 USC00319467                   WILMINGTON 7 N, NC US 2021-09-25 0.00   NA
## 2283 USC00319467                   WILMINGTON 7 N, NC US 2021-09-26 0.00   NA
## 2284 USC00319467                   WILMINGTON 7 N, NC US 2021-09-27 0.01   NA
## 2285 USC00319467                   WILMINGTON 7 N, NC US 2021-09-28 0.00   NA
## 2286 USC00319467                   WILMINGTON 7 N, NC US 2021-09-29 0.00   NA
## 2287 USC00319467                   WILMINGTON 7 N, NC US 2021-09-30 0.00   NA
## 2288 USC00319467                   WILMINGTON 7 N, NC US 2021-10-01 0.00   NA
## 2289 USC00319467                   WILMINGTON 7 N, NC US 2021-10-02 0.00   NA
## 2290 USC00319467                   WILMINGTON 7 N, NC US 2021-10-03 0.00   NA
## 2291 USC00319467                   WILMINGTON 7 N, NC US 2021-10-04 0.00   NA
## 2292 USC00319467                   WILMINGTON 7 N, NC US 2021-10-05 0.00   NA
## 2293 USC00319467                   WILMINGTON 7 N, NC US 2021-10-06 0.01   NA
## 2294 USC00319467                   WILMINGTON 7 N, NC US 2021-10-07 0.10   NA
## 2295 USC00319467                   WILMINGTON 7 N, NC US 2021-10-08 0.01   NA
## 2296 USC00319467                   WILMINGTON 7 N, NC US 2021-10-09 0.64   NA
## 2297 USC00319467                   WILMINGTON 7 N, NC US 2021-10-10 0.01   NA
## 2298 USC00319467                   WILMINGTON 7 N, NC US 2021-10-11 0.00   NA
## 2299 USC00319467                   WILMINGTON 7 N, NC US 2021-10-12 0.00   NA
## 2300 USC00319467                   WILMINGTON 7 N, NC US 2021-10-13 0.00   NA
## 2301 USC00319467                   WILMINGTON 7 N, NC US 2021-10-14 0.00   NA
## 2302 USC00319467                   WILMINGTON 7 N, NC US 2021-10-15 0.00   NA
## 2303 USC00319467                   WILMINGTON 7 N, NC US 2021-10-16 0.00   NA
## 2304 USC00319467                   WILMINGTON 7 N, NC US 2021-10-17 0.00   NA
## 2305 USC00319467                   WILMINGTON 7 N, NC US 2021-10-18 0.00   NA
## 2306 USC00319467                   WILMINGTON 7 N, NC US 2021-10-19 0.00   NA
## 2307 USC00319467                   WILMINGTON 7 N, NC US 2021-10-20 0.00   NA
## 2308 USC00319467                   WILMINGTON 7 N, NC US 2021-10-21 0.00   NA
## 2309 USC00319467                   WILMINGTON 7 N, NC US 2021-10-22 0.00   NA
## 2310 USC00319467                   WILMINGTON 7 N, NC US 2021-10-23 0.00   NA
## 2311 USC00319467                   WILMINGTON 7 N, NC US 2021-10-24 0.01   NA
## 2312 USC00319467                   WILMINGTON 7 N, NC US 2021-10-25 0.02   NA
## 2313 USC00319467                   WILMINGTON 7 N, NC US 2021-10-26 0.04   NA
## 2314 USC00319467                   WILMINGTON 7 N, NC US 2021-10-27 0.00   NA
## 2315 USC00319467                   WILMINGTON 7 N, NC US 2021-10-28 0.20   NA
## 2316 USC00319467                   WILMINGTON 7 N, NC US 2021-10-29 0.26   NA
## 2317 USC00319467                   WILMINGTON 7 N, NC US 2021-10-30 0.00   NA
## 2318 USC00319467                   WILMINGTON 7 N, NC US 2021-10-31 0.00   NA
## 2319 USC00319467                   WILMINGTON 7 N, NC US 2021-11-01 0.00   NA
## 2320 USC00319467                   WILMINGTON 7 N, NC US 2021-11-02 0.00   NA
## 2321 USC00319467                   WILMINGTON 7 N, NC US 2021-11-03 0.00   NA
## 2322 USC00319467                   WILMINGTON 7 N, NC US 2021-11-04 0.00   NA
## 2323 USC00319467                   WILMINGTON 7 N, NC US 2021-11-05 0.00   NA
## 2324 USC00319467                   WILMINGTON 7 N, NC US 2021-11-06 0.37   NA
## 2325 USC00319467                   WILMINGTON 7 N, NC US 2021-11-07 0.02   NA
## 2326 USC00319467                   WILMINGTON 7 N, NC US 2021-11-08 0.00   NA
## 2327 USC00319467                   WILMINGTON 7 N, NC US 2021-11-09 0.00   NA
## 2328 USC00319467                   WILMINGTON 7 N, NC US 2021-11-10 0.00   NA
## 2329 USC00319467                   WILMINGTON 7 N, NC US 2021-11-11 0.08   NA
## 2330 USC00319467                   WILMINGTON 7 N, NC US 2021-11-12 0.07   NA
## 2331 USC00319467                   WILMINGTON 7 N, NC US 2021-11-13 0.00   NA
## 2332 USC00319467                   WILMINGTON 7 N, NC US 2021-11-14 0.00   NA
## 2333 USC00319467                   WILMINGTON 7 N, NC US 2021-11-15 0.02   NA
## 2334 USC00319467                   WILMINGTON 7 N, NC US 2021-11-16 0.00   NA
## 2335 USC00319467                   WILMINGTON 7 N, NC US 2021-11-17 0.00   NA
## 2336 USC00319467                   WILMINGTON 7 N, NC US 2021-11-18 0.00   NA
## 2337 USC00319467                   WILMINGTON 7 N, NC US 2021-11-19 0.00   NA
## 2338 USC00319467                   WILMINGTON 7 N, NC US 2021-11-20 0.00   NA
## 2339 USC00319467                   WILMINGTON 7 N, NC US 2021-12-01 0.00   NA
## 2340 USC00319467                   WILMINGTON 7 N, NC US 2021-12-02 0.00   NA
## 2341 USC00319467                   WILMINGTON 7 N, NC US 2021-12-03 0.00   NA
## 2342 USC00319467                   WILMINGTON 7 N, NC US 2021-12-04 0.00   NA
## 2343 USC00319467                   WILMINGTON 7 N, NC US 2021-12-05 0.00   NA
## 2344 USC00319467                   WILMINGTON 7 N, NC US 2021-12-06 0.00   NA
## 2345 USC00319467                   WILMINGTON 7 N, NC US 2021-12-07 0.00   NA
## 2346 USC00319467                   WILMINGTON 7 N, NC US 2021-12-08 1.96   NA
## 2347 USC00319467                   WILMINGTON 7 N, NC US 2021-12-09 0.00   NA
## 2348 USC00319467                   WILMINGTON 7 N, NC US 2021-12-10 0.01   NA
## 2349 USC00319467                   WILMINGTON 7 N, NC US 2021-12-11 0.00   NA
## 2350 USC00319467                   WILMINGTON 7 N, NC US 2021-12-12 0.38   NA
## 2351 USC00319467                   WILMINGTON 7 N, NC US 2021-12-13 0.00   NA
## 2352 USC00319467                   WILMINGTON 7 N, NC US 2021-12-14 0.00   NA
## 2353 USC00319467                   WILMINGTON 7 N, NC US 2021-12-15 0.01   NA
## 2354 USC00319467                   WILMINGTON 7 N, NC US 2021-12-16 0.00   NA
## 2355 USC00319467                   WILMINGTON 7 N, NC US 2021-12-17 0.00   NA
## 2356 USC00319467                   WILMINGTON 7 N, NC US 2021-12-18 0.00   NA
## 2357 USC00319467                   WILMINGTON 7 N, NC US 2021-12-19 0.60   NA
## 2358 USC00319467                   WILMINGTON 7 N, NC US 2021-12-20 0.00   NA
## 2359 USC00319467                   WILMINGTON 7 N, NC US 2021-12-21 1.61   NA
## 2360 USC00319467                   WILMINGTON 7 N, NC US 2021-12-22 0.14   NA
## 2361 USC00319467                   WILMINGTON 7 N, NC US 2021-12-23 0.00   NA
## 2362 USC00319467                   WILMINGTON 7 N, NC US 2021-12-24 0.00   NA
## 2363 USC00319467                   WILMINGTON 7 N, NC US 2021-12-25 0.00   NA
## 2364 USC00319467                   WILMINGTON 7 N, NC US 2021-12-26 0.00   NA
## 2365 USC00319467                   WILMINGTON 7 N, NC US 2021-12-27 0.00   NA
## 2366 USC00319467                   WILMINGTON 7 N, NC US 2021-12-28 0.00   NA
## 2367 USC00319467                   WILMINGTON 7 N, NC US 2021-12-29 0.00   NA
## 2368 USC00319467                   WILMINGTON 7 N, NC US 2021-12-30 0.02   NA
## 2369 USC00319467                   WILMINGTON 7 N, NC US 2021-12-31 0.00   NA
## 2370 USC00319467                   WILMINGTON 7 N, NC US 2022-01-01 0.00   NA
## 2371 USC00319467                   WILMINGTON 7 N, NC US 2022-01-02 0.41   NA
## 2372 USC00319467                   WILMINGTON 7 N, NC US 2022-01-03 0.65   NA
## 2373 USC00319467                   WILMINGTON 7 N, NC US 2022-01-04 0.00   NA
## 2374 USC00319467                   WILMINGTON 7 N, NC US 2022-01-05 0.16   NA
## 2375 USC00319467                   WILMINGTON 7 N, NC US 2022-01-06 0.01   NA
## 2376 USC00319467                   WILMINGTON 7 N, NC US 2022-01-07 0.00   NA
## 2377 USC00319467                   WILMINGTON 7 N, NC US 2022-01-08 0.00   NA
## 2378 USC00319467                   WILMINGTON 7 N, NC US 2022-01-09 0.00   NA
## 2379 USC00319467                   WILMINGTON 7 N, NC US 2022-01-10 0.31   NA
## 2380 USC00319467                   WILMINGTON 7 N, NC US 2022-01-11 0.00   NA
## 2381 USC00319467                   WILMINGTON 7 N, NC US 2022-01-12 0.00   NA
## 2382 USC00319467                   WILMINGTON 7 N, NC US 2022-01-13 0.00   NA
## 2383 USC00319467                   WILMINGTON 7 N, NC US 2022-01-14 0.00   NA
## 2384 USC00319467                   WILMINGTON 7 N, NC US 2022-01-15 0.00   NA
## 2385 USC00319467                   WILMINGTON 7 N, NC US 2022-01-16 2.39   NA
## 2386 USC00319467                   WILMINGTON 7 N, NC US 2022-01-17 0.00   NA
## 2387 USC00319467                   WILMINGTON 7 N, NC US 2022-01-18 0.00   NA
## 2388 USC00319467                   WILMINGTON 7 N, NC US 2022-01-19 0.00   NA
## 2389 USC00319467                   WILMINGTON 7 N, NC US 2022-01-20 0.18   NA
## 2390 USC00319467                   WILMINGTON 7 N, NC US 2022-01-21 0.35   NA
## 2391 USC00319467                   WILMINGTON 7 N, NC US 2022-01-22 0.04   NA
## 2392 USC00319467                   WILMINGTON 7 N, NC US 2022-01-23 0.00   NA
## 2393 USC00319467                   WILMINGTON 7 N, NC US 2022-01-24 0.00   NA
## 2394 USC00319467                   WILMINGTON 7 N, NC US 2022-01-25 0.01   NA
## 2395 USC00319467                   WILMINGTON 7 N, NC US 2022-01-26 0.01   NA
## 2396 USC00319467                   WILMINGTON 7 N, NC US 2022-01-27 0.00   NA
## 2397 USC00319467                   WILMINGTON 7 N, NC US 2022-01-28 0.11   NA
## 2398 USC00319467                   WILMINGTON 7 N, NC US 2022-01-29 0.07   NA
## 2399 USC00319467                   WILMINGTON 7 N, NC US 2022-01-30 0.00   NA
## 2400 USC00319467                   WILMINGTON 7 N, NC US 2022-01-31 0.00   NA
## 2401 USC00319467                   WILMINGTON 7 N, NC US 2022-02-01 0.00   NA
## 2402 USC00319467                   WILMINGTON 7 N, NC US 2022-02-02 0.00   NA
## 2403 USC00319467                   WILMINGTON 7 N, NC US 2022-02-03 0.03   NA
## 2404 USC00319467                   WILMINGTON 7 N, NC US 2022-02-04 0.39   NA
## 2405 USC00319467                   WILMINGTON 7 N, NC US 2022-02-05 0.17   NA
## 2406 USC00319467                   WILMINGTON 7 N, NC US 2022-02-06 0.00   NA
## 2407 USC00319467                   WILMINGTON 7 N, NC US 2022-02-07 0.26   NA
## 2408 USC00319467                   WILMINGTON 7 N, NC US 2022-02-08 0.00   NA
## 2409 USC00319467                   WILMINGTON 7 N, NC US 2022-02-09 0.00   NA
## 2410 USC00319467                   WILMINGTON 7 N, NC US 2022-02-10 0.00   NA
## 2411 USC00319467                   WILMINGTON 7 N, NC US 2022-02-11 0.00   NA
## 2412 USC00319467                   WILMINGTON 7 N, NC US 2022-02-12 0.00   NA
## 2413 USC00319467                   WILMINGTON 7 N, NC US 2022-02-13 0.13   NA
## 2414 USC00319467                   WILMINGTON 7 N, NC US 2022-02-14 0.00   NA
## 2415 USC00319467                   WILMINGTON 7 N, NC US 2022-02-15 0.00   NA
## 2416 USC00319467                   WILMINGTON 7 N, NC US 2022-02-16 0.00   NA
## 2417 USC00319467                   WILMINGTON 7 N, NC US 2022-02-17 0.03   NA
## 2418 USC00319467                   WILMINGTON 7 N, NC US 2022-02-18 0.21   NA
## 2419 USC00319467                   WILMINGTON 7 N, NC US 2022-02-19 0.00   NA
## 2420 USC00319467                   WILMINGTON 7 N, NC US 2022-02-20 0.00   NA
## 2421 USC00319467                   WILMINGTON 7 N, NC US 2022-02-21 0.07   NA
## 2422 USC00319467                   WILMINGTON 7 N, NC US 2022-02-22 0.00   NA
## 2423 USC00319467                   WILMINGTON 7 N, NC US 2022-02-23 0.00   NA
## 2424 USC00319467                   WILMINGTON 7 N, NC US 2022-02-24 0.00   NA
## 2425 USC00319467                   WILMINGTON 7 N, NC US 2022-02-25 0.00   NA
## 2426 USC00319467                   WILMINGTON 7 N, NC US 2022-04-01 0.00   NA
## 2427 USC00319467                   WILMINGTON 7 N, NC US 2022-04-02 0.00   NA
## 2428 USC00319467                   WILMINGTON 7 N, NC US 2022-04-03 0.00   NA
## 2429 USC00319467                   WILMINGTON 7 N, NC US 2022-04-04 0.00   NA
## 2430 USC00319467                   WILMINGTON 7 N, NC US 2022-04-05 1.05   NA
## 2431 USC00319467                   WILMINGTON 7 N, NC US 2022-04-06 0.46   NA
## 2432 USC00319467                   WILMINGTON 7 N, NC US 2022-04-07 0.11   NA
## 2433 USC00319467                   WILMINGTON 7 N, NC US 2022-04-08 0.00   NA
## 2434 USC00319467                   WILMINGTON 7 N, NC US 2022-04-09 0.00   NA
## 2435 USC00319467                   WILMINGTON 7 N, NC US 2022-04-10 0.00   NA
## 2436 USC00319467                   WILMINGTON 7 N, NC US 2022-04-11 0.00   NA
## 2437 USC00319467                   WILMINGTON 7 N, NC US 2022-04-12 0.00   NA
## 2438 USC00319467                   WILMINGTON 7 N, NC US 2022-04-13 0.00   NA
## 2439 USC00319467                   WILMINGTON 7 N, NC US 2022-04-14 0.00   NA
## 2440 USC00319467                   WILMINGTON 7 N, NC US 2022-04-15 0.00   NA
## 2441 USC00319467                   WILMINGTON 7 N, NC US 2022-04-16 0.18   NA
## 2442 USC00319467                   WILMINGTON 7 N, NC US 2022-04-17 0.00   NA
## 2443 USC00319467                   WILMINGTON 7 N, NC US 2022-04-18 0.78   NA
## 2444 USC00319467                   WILMINGTON 7 N, NC US 2022-04-19 0.00   NA
## 2445 USC00319467                   WILMINGTON 7 N, NC US 2022-04-20 0.02   NA
## 2446 USC00319467                   WILMINGTON 7 N, NC US 2022-04-21 0.00   NA
## 2447 USC00319467                   WILMINGTON 7 N, NC US 2022-04-22 0.00   NA
## 2448 USC00319467                   WILMINGTON 7 N, NC US 2022-04-23 0.00   NA
## 2449 USC00319467                   WILMINGTON 7 N, NC US 2022-04-24 0.00   NA
## 2450 USC00319467                   WILMINGTON 7 N, NC US 2022-04-25 0.00   NA
## 2451 USC00319467                   WILMINGTON 7 N, NC US 2022-04-26 0.20   NA
## 2452 USC00319467                   WILMINGTON 7 N, NC US 2022-04-27 0.00   NA
## 2453 USC00319467                   WILMINGTON 7 N, NC US 2022-04-28 0.00   NA
## 2454 USC00319467                   WILMINGTON 7 N, NC US 2022-04-29 0.00   NA
## 2455 USC00319467                   WILMINGTON 7 N, NC US 2022-04-30 0.00   NA
## 2456 USC00319467                   WILMINGTON 7 N, NC US 2022-05-01 0.00   NA
## 2457 USC00319467                   WILMINGTON 7 N, NC US 2022-05-02 0.00   NA
## 2458 USC00319467                   WILMINGTON 7 N, NC US 2022-05-03 0.00   NA
## 2459 USC00319467                   WILMINGTON 7 N, NC US 2022-05-04 0.00   NA
## 2460 USC00319467                   WILMINGTON 7 N, NC US 2022-05-05 0.00   NA
## 2461 USC00319467                   WILMINGTON 7 N, NC US 2022-05-06 0.02   NA
## 2462 USC00319467                   WILMINGTON 7 N, NC US 2022-05-07 0.03   NA
## 2463 USC00319467                   WILMINGTON 7 N, NC US 2022-05-08 0.00   NA
## 2464 USC00319467                   WILMINGTON 7 N, NC US 2022-05-09 0.00   NA
## 2465 USC00319467                   WILMINGTON 7 N, NC US 2022-05-10 0.00   NA
## 2466 USC00319467                   WILMINGTON 7 N, NC US 2022-05-11 0.00   NA
## 2467 USC00319467                   WILMINGTON 7 N, NC US 2022-05-12 0.29   NA
## 2468 USC00319467                   WILMINGTON 7 N, NC US 2022-05-13 0.29   NA
## 2469 USC00319467                   WILMINGTON 7 N, NC US 2022-05-14 0.24   NA
## 2470 USC00319467                   WILMINGTON 7 N, NC US 2022-05-15 0.00   NA
## 6797 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-04 0.00   NA
## 6798 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-05 0.09   NA
## 6799 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-06 0.00   NA
## 6800 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-07 0.00   NA
## 6801 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-08 0.72   NA
## 6802 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-09 0.00   NA
## 6803 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-10 0.00   NA
## 6804 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-11 0.18   NA
## 6805 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-12 0.46   NA
## 6806 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-13 0.00   NA
## 6807 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-14 0.12   NA
## 6808 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-15 0.20   NA
## 6809 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-16 0.00   NA
## 6810 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-17 0.00   NA
## 6811 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-18 0.00   NA
## 6812 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-19 0.00   NA
## 6813 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-20 0.00   NA
## 6814 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-21 0.01   NA
## 6815 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-22 0.00   NA
## 6816 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-23 0.00   NA
## 6817 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-24 0.00   NA
## 6818 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-25 0.40   NA
## 6819 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-26 0.06   NA
## 6820 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-27 0.30   NA
## 6821 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-28 0.24   NA
## 6822 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-29 0.00   NA
## 6823 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-30 0.00   NA
## 6824 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-01-31 1.44   NA
## 6825 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-01 0.01   NA
## 6826 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-02 0.00   NA
## 6827 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-03 0.00   NA
## 6828 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-04 0.00   NA
## 6829 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-05 0.09   NA
## 6830 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-06 0.10   NA
## 6831 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-07 0.62   NA
## 6832 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-08 0.00   NA
## 6833 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-09 0.08   NA
## 6834 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-10 0.07   NA
## 6835 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-11 0.05   NA
## 6836 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-12 0.71   NA
## 6837 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-13 0.83   NA
## 6838 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-14 1.25   NA
## 6839 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-15 0.03   NA
## 6840 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-16 0.13   NA
## 6841 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-17 0.00   NA
## 6842 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-18 0.67   NA
## 6843 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-19 1.10   NA
## 6844 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-20 0.00   NA
## 6845 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-21 0.00   NA
## 6846 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-22 0.26   NA
## 6847 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-23 0.00   NA
## 6848 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-24 0.00   NA
## 6849 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-25 0.00   NA
## 6850 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-26 0.04   NA
## 6851 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-27 0.00   NA
## 6852 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-02-28 0.00   NA
## 6853 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-01 0.03   NA
## 6854 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-02 0.01   NA
## 6855 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-03 0.18   NA
## 6856 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-04 0.00   NA
## 6857 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-05 0.00   NA
## 6858 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-06 0.00   NA
## 6859 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-07 0.00   NA
## 6860 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-08 0.00   NA
## 6861 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-09 0.00   NA
## 6862 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-10 0.00   NA
## 6863 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-11 0.00   NA
## 6864 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-12 0.00   NA
## 6865 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-13 0.00   NA
## 6866 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-14 0.00   NA
## 6867 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-15 0.00   NA
## 6868 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-16 0.16   NA
## 6869 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-17 0.54   NA
## 6870 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-18 0.00   NA
## 6871 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-19 0.00   NA
## 6872 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-20 0.10   NA
## 6873 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-21 0.00   NA
## 6874 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-22 0.01   NA
## 6875 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-23 0.05   NA
## 6876 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-24 0.06   NA
## 6877 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-25 0.00   NA
## 6878 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-26 0.00   NA
## 6879 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-27 0.00   NA
## 6880 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-28 0.16   NA
## 6881 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-29 0.00   NA
## 6882 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-30 0.00   NA
## 6883 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-03-31 1.32   NA
## 6884 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-01 0.00   NA
## 6885 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-02 0.00   NA
## 6886 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-03 0.00   NA
## 6887 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-04 0.00   NA
## 6888 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-05 0.00   NA
## 6889 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-06 0.00   NA
## 6890 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-07 0.00   NA
## 6891 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-08 0.00   NA
## 6892 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-09 0.00   NA
## 6893 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-10 0.13   NA
## 6894 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-11 0.40   NA
## 6895 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-12 0.00   NA
## 6896 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-13 0.00   NA
## 6897 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-14 0.00   NA
## 6898 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-15 0.10   NA
## 6899 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-16 0.00   NA
## 6900 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-17 0.00   NA
## 6901 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-18 0.00   NA
## 6902 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-19 0.00   NA
## 6903 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-20 0.00   NA
## 6904 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-21 0.00   NA
## 6905 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-22 0.00   NA
## 6906 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-23 0.00   NA
## 6907 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-24 0.00   NA
## 6908 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-25 0.11   NA
## 6909 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-26 0.00   NA
## 6910 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-27 0.00   NA
## 6911 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-28 0.00   NA
## 6912 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-29 0.00   NA
## 6913 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-04-30 0.00   NA
## 6914 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-01 0.00   NA
## 6915 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-02 0.00   NA
## 6916 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-03 0.02   NA
## 6917 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-04 0.00   NA
## 6918 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-05 0.00   NA
## 6919 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-06 0.00   NA
## 6920 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-07 0.44   NA
## 6921 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-08 0.00   NA
## 6922 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-09 0.00   NA
## 6923 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-10 0.00   NA
## 6924 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-11 0.00   NA
## 6925 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-12 0.29   NA
## 6926 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-13 0.00   NA
## 6927 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-14 0.00   NA
## 6928 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-15 0.00   NA
## 6929 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-16 0.00   NA
## 6930 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-17 0.00   NA
## 6931 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-18 0.00   NA
## 6932 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-19 0.00   NA
## 6933 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-20 0.00   NA
## 6934 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-21 0.00   NA
## 6935 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-22 0.00   NA
## 6936 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-23 0.00   NA
## 6937 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-24 0.02   NA
## 6938 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-25 0.03   NA
## 6939 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-26 0.00   NA
## 6940 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-27 0.00   NA
## 6941 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-28 0.00   NA
## 6942 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-29 0.00   NA
## 6943 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-30 0.15   NA
## 6944 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-05-31 0.00   NA
## 6945 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-01 0.00   NA
## 6946 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-02 0.82   NA
## 6947 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-03 0.87   NA
## 6948 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-04 1.82   NA
## 6949 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-05 0.00   NA
## 6950 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-06 0.02   NA
## 6951 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-07 0.00   NA
## 6952 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-08 0.00   NA
## 6953 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-09 0.04   NA
## 6954 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-10 0.41   NA
## 6955 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-11 0.36   NA
## 6956 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-12 2.82   NA
## 6957 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-13 0.03   NA
## 6958 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-14 0.00   NA
## 6959 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-15 1.65   NA
## 6960 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-16 0.00   NA
## 6961 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-17 0.00   NA
## 6962 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-18 0.00   NA
## 6963 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-19 0.00   NA
## 6964 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-20 1.23   NA
## 6965 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-21 0.10   NA
## 6966 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-22 0.34   NA
## 6967 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-23 0.00   NA
## 6968 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-24 0.00   NA
## 6969 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-25 1.05   NA
## 6970 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-26 0.63   NA
## 6971 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-27 0.00   NA
## 6972 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-28 0.06   NA
## 6973 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-29 0.01   NA
## 6974 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-06-30 0.00   NA
## 6975 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-01 0.00   NA
## 6976 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-02 1.27   NA
## 6977 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-03 0.00   NA
## 6978 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-04 0.00   NA
## 6979 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-05 0.00   NA
## 6980 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-06 0.00   NA
## 6981 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-07 0.00   NA
## 6982 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-08 0.52   NA
## 6983 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-09 0.73   NA
## 6984 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-10 0.00   NA
## 6985 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-11 0.05   NA
## 6986 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-12 0.01   NA
## 6987 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-13 0.41   NA
## 6988 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-14 1.50   NA
## 6989 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-15 0.01   NA
## 6990 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-16 0.00   NA
## 6991 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-17 0.00   NA
## 6992 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-18 1.26   NA
## 6993 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-19 2.49   NA
## 6994 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-20 0.12   NA
## 6995 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-21 0.00   NA
## 6996 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-22 0.02   NA
## 6997 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-23 0.00   NA
## 6998 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-24 0.00   NA
## 6999 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-25 0.00   NA
## 7000 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-26 0.07   NA
## 7001 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-27 0.00   NA
## 7002 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-28 0.23   NA
## 7003 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-29 0.00   NA
## 7004 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-30 0.00   NA
## 7005 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-07-31 0.16   NA
## 7006 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-01 0.32   NA
## 7007 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-02 0.55   NA
## 7008 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-03 2.97   NA
## 7009 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-04 0.23   NA
## 7010 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-05 0.00   NA
## 7011 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-06 1.39   NA
## 7012 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-07 0.90   NA
## 7013 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-08 0.02   NA
## 7014 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-09 0.00   NA
## 7015 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-10 0.00   NA
## 7016 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-11 0.00   NA
## 7017 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-12 0.00   NA
## 7018 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-13 0.00   NA
## 7019 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-14 0.00   NA
## 7020 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-15 0.03   NA
## 7021 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-16 0.35   NA
## 7022 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-17 1.02   NA
## 7023 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-18 0.87   NA
## 7024 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-19 0.00   NA
## 7025 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-20 1.10   NA
## 7026 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-21 0.30   NA
## 7027 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-22 0.00   NA
## 7028 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-23 0.00   NA
## 7029 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-24 0.00   NA
## 7030 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-25 0.02   NA
## 7031 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-26 0.00   NA
## 7032 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-27 0.00   NA
## 7033 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-28 0.00   NA
## 7034 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-29 0.00   NA
## 7035 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-30 0.00   NA
## 7036 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-08-31 0.00   NA
## 7037 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-01 0.47   NA
## 7038 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-02 0.00   NA
## 7039 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-03 0.00   NA
## 7040 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-04 0.00   NA
## 7041 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-05 0.00   NA
## 7042 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-06 0.04   NA
## 7043 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-07 0.46   NA
## 7044 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-08 0.04   NA
## 7045 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-09 0.33   NA
## 7046 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-10 0.00   NA
## 7047 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-11 0.00   NA
## 7048 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-12 0.00   NA
## 7049 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-13 0.00   NA
## 7050 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-14 0.00   NA
## 7051 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-15 0.00   NA
## 7052 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-16 0.00   NA
## 7053 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-17 0.00   NA
## 7054 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-18 0.00   NA
## 7055 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-19 0.00   NA
## 7056 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-20 1.14   NA
## 7057 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-21 3.79   NA
## 7058 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-22 4.31   NA
## 7059 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-23 0.19   NA
## 7060 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-24 0.00   NA
## 7061 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-25 0.00   NA
## 7062 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-26 0.00   NA
## 7063 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-27 0.00   NA
## 7064 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-28 0.00   NA
## 7065 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-29 0.00   NA
## 7066 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-09-30 0.00   NA
## 7067 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-01 0.00   NA
## 7068 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-02 0.00   NA
## 7069 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-03 0.00   NA
## 7070 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-04 0.03   NA
## 7071 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-05 0.00   NA
## 7072 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-06 0.02   NA
## 7073 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-07 0.08   NA
## 7074 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-08 0.00   NA
## 7075 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-09 0.49   NA
## 7076 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-10 0.00   NA
## 7077 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-11 0.00   NA
## 7078 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-12 0.00   NA
## 7079 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-13 0.00   NA
## 7080 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-14 0.00   NA
## 7081 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-15 0.00   NA
## 7082 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-16 0.01   NA
## 7083 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-17 0.00   NA
## 7084 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-18 0.00   NA
## 7085 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-19 0.00   NA
## 7086 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-20 0.00   NA
## 7087 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-21 0.00   NA
## 7088 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-22 0.00   NA
## 7089 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-23 0.00   NA
## 7090 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-24 0.00   NA
## 7091 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-25 0.04   NA
## 7092 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-26 0.11   NA
## 7093 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-27 0.00   NA
## 7094 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-28 0.14   NA
## 7095 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-29 0.22   NA
## 7096 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-30 0.00   NA
## 7097 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-10-31 0.00   NA
## 7098 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-01 0.00   NA
## 7099 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-02 0.00   NA
## 7100 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-03 0.00   NA
## 7101 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-04 0.00   NA
## 7102 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-05 0.00   NA
## 7103 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-06 0.34   NA
## 7104 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-07 0.00   NA
## 7105 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-08 0.00   NA
## 7106 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-09 0.00   NA
## 7107 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-10 0.00   NA
## 7108 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-11 0.05   NA
## 7109 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-12 0.09   NA
## 7110 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-13 0.00   NA
## 7111 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-14 0.00   NA
## 7112 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-15 0.00   NA
## 7113 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-16 0.00   NA
## 7114 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-17 0.00   NA
## 7115 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-18 0.00   NA
## 7116 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-19 0.00   NA
## 7117 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-20 0.00   NA
## 7118 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-21 0.00   NA
## 7119 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-22 0.07   NA
## 7120 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-23 0.00   NA
## 7121 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-24 0.00   NA
## 7122 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-25 0.00   NA
## 7123 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-26 0.18   NA
## 7124 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-27 0.00   NA
## 7125 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-28 0.00   NA
## 7126 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-29 0.00   NA
## 7127 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-11-30 0.00   NA
## 7128 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-01 0.00   NA
## 7129 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-02 0.00   NA
## 7130 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-03 0.00   NA
## 7131 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-04 0.00   NA
## 7132 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-05 0.00   NA
## 7133 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-06 0.00   NA
## 7134 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-07 0.00   NA
## 7135 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-08 0.96   NA
## 7136 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-09 0.00   NA
## 7137 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-10 0.06   NA
## 7138 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-11 0.00   NA
## 7139 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-12 0.30   NA
## 7140 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-13 0.00   NA
## 7141 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-14 0.00   NA
## 7142 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-15 0.00   NA
## 7143 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-16 0.00   NA
## 7144 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-17 0.00   NA
## 7145 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-18 0.00   NA
## 7146 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-19 0.13   NA
## 7147 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-20 0.00   NA
## 7148 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-21 0.94   NA
## 7149 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-22 0.11   NA
## 7150 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-23 0.00   NA
## 7151 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-24 0.00   NA
## 7152 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-25 0.00   NA
## 7153 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-26 0.00   NA
## 7154 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-27 0.00   NA
## 7155 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-28 0.00   NA
## 7156 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-29 0.00   NA
## 7157 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-30 0.03   NA
## 7158 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2021-12-31 0.00   NA
## 7159 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-01 0.00   NA
## 7160 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-02 0.34   NA
## 7161 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-03 0.98   NA
## 7162 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-04 0.00   NA
## 7163 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-05 0.07   NA
## 7164 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-06 0.00   NA
## 7165 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-07 0.00   NA
## 7166 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-08 0.00   NA
## 7167 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-09 0.00   NA
## 7168 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-10 0.21   NA
## 7169 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-11 0.00   NA
## 7170 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-12 0.00   NA
## 7171 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-13 0.00   NA
## 7172 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-14 0.00   NA
## 7173 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-15 0.00   NA
## 7174 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-16 2.06   NA
## 7175 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-17 0.00   NA
## 7176 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-18 0.00   NA
## 7177 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-19 0.00   NA
## 7178 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-20 0.09   NA
## 7179 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-21 0.15   NA
## 7180 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-22 0.14   NA
## 7181 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-23 0.00   NA
## 7182 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-24 0.00   NA
## 7183 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-25 0.02   NA
## 7184 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-26 0.00   NA
## 7185 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-27 0.00   NA
## 7186 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-28 0.03   NA
## 7187 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-29 0.05   NA
## 7188 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-30 0.00   NA
## 7189 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-01-31 0.00   NA
## 7190 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-01 0.00   NA
## 7191 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-02 0.01   NA
## 7192 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-03 0.02   NA
## 7193 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-04 0.34   NA
## 7194 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-05 0.07   NA
## 7195 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-06 0.00   NA
## 7196 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-07 0.26   NA
## 7197 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-08 0.00   NA
## 7198 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-09 0.00   NA
## 7199 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-10 0.00   NA
## 7200 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-11 0.00   NA
## 7201 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-12 0.00   NA
## 7202 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-13 0.05   NA
## 7203 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-14 0.00   NA
## 7204 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-15 0.00   NA
## 7205 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-16 0.00   NA
## 7206 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-17 0.07   NA
## 7207 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-18 0.14   NA
## 7208 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-19 0.00   NA
## 7209 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-20 0.00   NA
## 7210 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-21 0.06   NA
## 7211 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-22 0.00   NA
## 7212 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-23 0.00   NA
## 7213 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-24 0.00   NA
## 7214 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-25 0.00   NA
## 7215 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-26 0.00   NA
## 7216 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-27 0.35   NA
## 7217 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-02-28 0.00   NA
## 7218 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-01 0.00   NA
## 7219 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-02 0.00   NA
## 7220 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-03 0.00   NA
## 7221 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-04 0.00   NA
## 7222 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-05 0.00   NA
## 7223 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-06 0.00   NA
## 7224 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-07 0.00   NA
## 7225 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-08 0.00   NA
## 7226 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-09 0.28   NA
## 7227 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-10 0.03   NA
## 7228 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-11 0.05   NA
## 7229 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-12 0.22   NA
## 7230 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-13 0.00   NA
## 7231 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-14 0.00   NA
## 7232 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-15 0.00   NA
## 7233 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-16 0.11   NA
## 7234 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-17 0.00   NA
## 7235 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-18 0.00   NA
## 7236 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-19 0.00   NA
## 7237 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-20 0.00   NA
## 7238 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-21 0.00   NA
## 7239 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-22 0.00   NA
## 7240 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-23 0.04   NA
## 7241 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-24 1.41   NA
## 7242 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-25 0.14   NA
## 7243 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-26 0.00   NA
## 7244 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-27 0.00   NA
## 7245 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-28 0.00   NA
## 7246 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-29 0.00   NA
## 7247 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-30 0.02   NA
## 7248 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-03-31 0.13   NA
## 7249 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-01 0.00   NA
## 7250 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-02 0.00   NA
## 7251 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-03 0.00   NA
## 7252 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-04 0.00   NA
## 7253 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-05 1.03   NA
## 7254 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-06 0.53   NA
## 7255 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-07 0.04   NA
## 7256 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-08 0.00   NA
## 7257 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-09 0.00   NA
## 7258 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-10 0.00   NA
## 7259 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-11 0.00   NA
## 7260 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-12 0.00   NA
## 7261 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-13 0.00   NA
## 7262 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-14 0.00   NA
## 7263 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-15 0.00   NA
## 7264 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-16 0.37   NA
## 7265 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-17 0.00   NA
## 7266 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-18 1.20   NA
## 7267 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-19 0.00   NA
## 7268 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-20 0.00   NA
## 7269 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-21 0.00   NA
## 7270 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-22 0.00   NA
## 7271 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-23 0.00   NA
## 7272 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-24 0.00   NA
## 7273 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-25 0.00   NA
## 7274 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-26 0.10   NA
## 7275 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-27 0.00   NA
## 7276 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-28 0.00   NA
## 7277 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-29 0.00   NA
## 7278 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-04-30 0.00   NA
## 7279 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-01 0.00   NA
## 7280 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-02 0.00   NA
## 7281 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-03 0.00   NA
## 7282 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-04 0.00   NA
## 7283 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-05 0.03   NA
## 7284 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-06 0.00   NA
## 7285 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-07 0.00   NA
## 7286 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-08 0.00   NA
## 7287 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-09 0.00   NA
## 7288 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-10 0.00   NA
## 7289 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-11 0.00   NA
## 7290 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-12 0.21   NA
## 7291 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-13 0.12   NA
## 7292 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-14 0.00   NA
## 7293 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-15 0.00   NA
## 7294 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-16 0.31   NA
## 7295 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-17 0.02   NA
## 7296 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-18 0.00   NA
## 7297 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-19 0.00   NA
## 7298 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-20 0.00   NA
## 7299 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-21 0.00   NA
## 7300 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-22 0.02   NA
## 7301 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-23 0.00   NA
## 7302 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-24 0.00   NA
## 7303 USW00013748 WILMINGTON INTERNATIONAL AIRPORT, NC US 2022-05-25 0.00   NA
##      TMAX TMIN TOBS
## 1      68   47   47
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## 7193   75   63   NA
## 7194   63   32   NA
## 7195   55   32   NA
## 7196   52   41   NA
## 7197   52   33   NA
## 7198   61   28   NA
## 7199   66   34   NA
## 7200   70   37   NA
## 7201   71   41   NA
## 7202   59   39   NA
## 7203   54   30   NA
## 7204   52   28   NA
## 7205   67   34   NA
## 7206   75   46   NA
## 7207   76   47   NA
## 7208   61   38   NA
## 7209   54   30   NA
## 7210   71   39   NA
## 7211   74   57   NA
## 7212   82   61   NA
## 7213   65   54   NA
## 7214   81   55   NA
## 7215   64   43   NA
## 7216   49   40   NA
## 7217   60   38   NA
## 7218   69   39   NA
## 7219   77   42   NA
## 7220   84   48   NA
## 7221   64   46   NA
## 7222   74   48   NA
## 7223   79   62   NA
## 7224   80   66   NA
## 7225   72   55   NA
## 7226   78   50   NA
## 7227   57   47   NA
## 7228   57   47   NA
## 7229   73   32   NA
## 7230   50   25   NA
## 7231   64   31   NA
## 7232   70   39   NA
## 7233   71   56   NA
## 7234   77   57   NA
## 7235   76   47   NA
## 7236   82   65   NA
## 7237   70   48   NA
## 7238   68   37   NA
## 7239   75   40   NA
## 7240   78   55   NA
## 7241   73   60   NA
## 7242   72   54   NA
## 7243   69   47   NA
## 7244   66   40   NA
## 7245   65   34   NA
## 7246   59   35   NA
## 7247   72   45   NA
## 7248   80   67   NA
## 7249   79   52   NA
## 7250   65   42   NA
## 7251   77   48   NA
## 7252   64   44   NA
## 7253   79   51   NA
## 7254   83   64   NA
## 7255   81   62   NA
## 7256   73   48   NA
## 7257   62   46   NA
## 7258   67   41   NA
## 7259   80   43   NA
## 7260   83   62   NA
## 7261   82   61   NA
## 7262   82   65   NA
## 7263   74   51   NA
## 7264   75   51   NA
## 7265   78   57   NA
## 7266   68   50   NA
## 7267   65   45   NA
## 7268   63   40   NA
## 7269   72   45   NA
## 7270   79   51   NA
## 7271   79   55   NA
## 7272   82   56   NA
## 7273   84   57   NA
## 7274   86   64   NA
## 7275   75   54   NA
## 7276   72   48   NA
## 7277   72   48   NA
## 7278   81   59   NA
## 7279   83   62   NA
## 7280   87   67   NA
## 7281   88   70   NA
## 7282   86   71   NA
## 7283   84   69   NA
## 7284   88   69   NA
## 7285   83   62   NA
## 7286   62   51   NA
## 7287   73   47   NA
## 7288   75   51   NA
## 7289   78   57   NA
## 7290   77   62   NA
## 7291   77   64   NA
## 7292   79   62   NA
## 7293   87   62   NA
## 7294   87   69   NA
## 7295   84   63   NA
## 7296   87   63   NA
## 7297   94   72   NA
## 7298   95   76   NA
## 7299   91   77   NA
## 7300   87   73   NA
## 7301   90   72   NA
## 7302   88   71   NA
## 7303   83   66   NA
full_temp_data_new_hanover[238,]
##         STATION                   NAME       DATE PRCP TAVG TMAX TMIN TOBS
## 239 USC00319461 WILMINGTON 7 SE, NC US 2021-08-31    0   NA   94   74   76
full_temp_data_new_hanover[237,]
##         STATION                   NAME       DATE PRCP TAVG TMAX TMIN TOBS
## 237 USC00319461 WILMINGTON 7 SE, NC US 2021-08-29    0   NA   88   64   76
full_temp_data_new_hanover[239,]
##         STATION                   NAME       DATE PRCP TAVG TMAX TMIN TOBS
## 240 USC00319461 WILMINGTON 7 SE, NC US 2021-09-01 0.04   NA   90   76   79
full_temp_data_new_hanover <- full_temp_data_new_hanover %>%
  rowwise() %>% 
  mutate(TMED = median(c(TMIN:TMAX)))

table(full_temp_data_new_hanover$STATION)
## 
## USC00319461 USC00319467 USW00013748 
##         505         412         507
hanover_temp <- full_temp_data_new_hanover %>% group_by(DATE) %>% summarise(mean_temp = mean(TMED))

hanover_weather <- merge(hanover_weather_prcp,hanover_temp, order.by=DATE)
hanover_weather$DATE <- as.Date(hanover_weather$DATE)
summary(hanover_weather)
##       DATE            mean_precipation    mean_temp    
##  Min.   :2021-01-04   Min.   :0.00000   Min.   :30.00  
##  1st Qu.:2021-05-10   1st Qu.:0.00000   1st Qu.:51.25  
##  Median :2021-09-14   Median :0.01057   Median :62.50  
##  Mean   :2021-09-14   Mean   :0.15017   Mean   :62.64  
##  3rd Qu.:2022-01-18   3rd Qu.:0.10680   3rd Qu.:74.71  
##  Max.   :2022-05-25   Max.   :3.82208   Max.   :85.83
colnames(hanover_weather)[1]<-"Date"
summary(hanover_weather)
##       Date            mean_precipation    mean_temp    
##  Min.   :2021-01-04   Min.   :0.00000   Min.   :30.00  
##  1st Qu.:2021-05-10   1st Qu.:0.00000   1st Qu.:51.25  
##  Median :2021-09-14   Median :0.01057   Median :62.50  
##  Mean   :2021-09-14   Mean   :0.15017   Mean   :62.64  
##  3rd Qu.:2022-01-18   3rd Qu.:0.10680   3rd Qu.:74.71  
##  Max.   :2022-05-25   Max.   :3.82208   Max.   :85.83
#precipation and temp plots#

wake_prep_plot <- wake_weather %>% ggplot(aes(Date,mean_precipation)) + 
  geom_line() + xlab("") + ylab("") + theme_bw(base_size = 14)

meck_prep_plot <- meck_weather %>% ggplot(aes(Date,mean_precipation)) + 
  geom_line() + xlab("") + ylab("") + theme_bw(base_size = 14)

hanover_prep_plot <- hanover_weather %>% ggplot(aes(Date,mean_precipation)) + 
  geom_line() + ylab("") + theme_bw(base_size = 14)

png(filename="precipitation.png", res = 500,units = "cm", width = 20, height = 10)
grid.arrange(wake_prep_plot,meck_prep_plot,hanover_prep_plot,
             left = text_grob("Mean Precipitaion (in inches)", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2
wake_temp_plot <- wake_weather %>% ggplot(aes(Date,TAVG)) + 
  geom_line() + xlab("") + ylab("") + theme_bw(base_size = 14)

meck_temp_plot <- meck_weather %>% ggplot(aes(Date,TAVG)) + 
  geom_line() + xlab("") + ylab("") + theme_bw(base_size = 14)

hanover_temp_plot <- hanover_weather %>% ggplot(aes(Date,mean_temp)) + 
  geom_line() + ylab("") + theme_bw(base_size = 14)

png(filename="temp.png", res = 500,units = "cm", width = 20, height = 10)
grid.arrange(wake_temp_plot,meck_temp_plot,hanover_temp_plot,
             left = text_grob("Average Temperature (in °F)", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2
#merge datasets

full_cases_wastewater_weather_data <- merge(full_cases_wastewater_data,
                                            wake_weather, order.by=Date)
full_cases_wastewater_weather_data_meck <- merge(full_cases_wastewater_data_meck,
                                            meck_weather, order.by=Date)
full_cases_wastewater_weather_data_hanover <- merge(full_cases_wastewater_data_hanover,
                                                 hanover_weather, order.by=Date)

#Correlations

png(filename="wake_correlations.png", 
    res = 700,units = "cm", width = 20, height = 10)
ggpairs(full_cases_wastewater_weather_data, 
        columns = 2:5,
        columnLabels = c("New Cases","Viral Gene",
                         "Precipitation", "Temperature")) +
  theme_bw()
dev.off()
## quartz_off_screen 
##                 2
png(filename="meck_correlations.png", 
    res = 700,units = "cm", width = 20, height = 10)
ggpairs(full_cases_wastewater_weather_data_meck, 
        columns = c(2,4:6),
        columnLabels = c("New Cases","Viral Gene",
                         "Precipitation", "Temperature")) +
  theme_bw()
dev.off()
## quartz_off_screen 
##                 2
png(filename="hanover_correlations.png", 
    res = 700,units = "cm", width = 20, height = 10)
ggpairs(full_cases_wastewater_weather_data_hanover, 
        columns = c(2:5),
        columnLabels = c("New Cases","Viral Gene",
                         "Precipitation", "Temperature")) +
  theme_bw()
dev.off()
## quartz_off_screen 
##                 2

Modelling COVID-19 cases only

ARIMA modelling

### Wake
df_wake_1 <- xts(df_wake$mean_new_cases,order.by = df_wake$Date)
df_wake_1 <- df_wake_1[-c(1,506,507,508),] #making even weekly data
attr(df_wake_1, 'frequency') <- 7 
periodicity(df_wake_1)  
## Daily periodicity from 2021-01-04 to 2022-05-22
df_wake_1_ts <- as.ts(df_wake_1)
plot(decompose(log(df_wake_1_ts))) #exponential growth is evident at the peakk

df_wake_seasonal_decomp <- decompose(log(df_wake_1_ts))

png(filename = "Additive_season.png",res = 700, units = "cm",width = 20, height = 14)
plot(df_wake_seasonal_decomp)
dev.off()
## quartz_off_screen 
##                 2
df_wake_deseasonal_decomp <- seasadj(df_wake_seasonal_decomp)

png(filename = "season_adjust.png",res = 700, units = "cm",width = 20, height = 12)
tsdisplay(df_wake_deseasonal_decomp, main = NULL)
dev.off()
## quartz_off_screen 
##                 2
#Forecasting

train_df_seasonal <- ts(log(df_wake_1_ts)[-c(491:504)], frequency=7)
test_df_seasonal <- log(df_wake_1_ts)[c(491:504)]

train_df1 <- df_wake_deseasonal_decomp[-c(491:504)]
test_df1 <- df_wake_deseasonal_decomp[c(491:504)]

lowest_rmse<-Inf
lowest_mae<-Inf
best_mod<-NULL
best_mod_mae<-NULL

for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    rmse_mod_1 <- rmse(test_df1,forecast_fit$mean)
    if (rmse_mod_1 < lowest_rmse){
      lowest_rmse <- rmse_mod_1 
      best_mod <- arima_mod_1
    }
  }
} #arima(3,1,4) gives the lowest RMSE

lowest_rmse
## [1] 0.1988205
best_mod
## Series: train_df1 
## ARIMA(3,1,4) 
## 
## Coefficients:
##          ar1      ar2     ar3      ma1     ma2      ma3     ma4
##       1.6674  -1.3488  0.6348  -2.1967  2.1290  -1.2446  0.3569
## s.e.  0.1944   0.2716  0.1394   0.1979  0.3862   0.3110  0.1147
## 
## sigma^2 = 0.1107:  log likelihood = -152.62
## AIC=321.24   AICc=321.54   BIC=354.77
for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    mae_mod <- mae(test_df1,forecast_fit$mean)
    if (mae_mod < lowest_mae){
      lowest_mae <- mae_mod
      best_mod_mae <- arima_mod_1
    }
  }
} #arima(3,1,4) gives the lowest RMSE

best_mod_mae
## Series: train_df1 
## ARIMA(3,1,4) 
## 
## Coefficients:
##          ar1      ar2     ar3      ma1     ma2      ma3     ma4
##       1.6674  -1.3488  0.6348  -2.1967  2.1290  -1.2446  0.3569
## s.e.  0.1944   0.2716  0.1394   0.1979  0.3862   0.3110  0.1147
## 
## sigma^2 = 0.1107:  log likelihood = -152.62
## AIC=321.24   AICc=321.54   BIC=354.77
lowest_mae
## [1] 0.170638
wake_fit_1_arima <- Arima(train_df1, order = c(4,1,4))
coeftest(wake_fit_1_arima)
## 
## z test of coefficients:
## 
##     Estimate Std. Error z value Pr(>|z|)   
## ar1 -0.24418    0.34159 -0.7148 0.474722   
## ar2  0.27374    0.12972  2.1103 0.034835 * 
## ar3 -0.53331    0.20916 -2.5498 0.010779 * 
## ar4 -0.16937    0.13204 -1.2827 0.199593   
## ma1 -0.23533    0.34260 -0.6869 0.492137   
## ma2 -0.51443    0.18590 -2.7673 0.005653 **
## ma3  0.61676    0.25956  2.3762 0.017491 * 
## ma4 -0.03240    0.22251 -0.1456 0.884229   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
wake_fit_1_arima_forecast <- forecast::forecast(wake_fit_1_arima,h=14)
rmse(test_df1,wake_fit_1_arima_forecast$mean)
## [1] 0.2931432
mae(test_df1,wake_fit_1_arima_forecast$mean) 
## [1] 0.2698372
checkresiduals(wake_fit_1_arima) 

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(4,1,4)
## Q* = 13.984, df = 3, p-value = 0.002928
## 
## Model df: 8.   Total lags used: 11
exp(wake_fit_1_arima_forecast$mean[1])
## [1] 5.387371
exp(wake_fit_1_arima_forecast$lower[1,])
##      80%      95% 
## 3.500936 2.786701
exp(wake_fit_1_arima_forecast$upper[1,])
##       80%       95% 
##  8.290289 10.415101
exp(wake_fit_1_arima_forecast$mean[1])-exp(test_df1[1])
## [1] -3.330173
exp(test_df1[7])
## [1] 6.998252
exp(wake_fit_1_arima_forecast$mean[7])
## [1] 5.625453
exp(wake_fit_1_arima_forecast$lower[7,])
##      80%      95% 
## 2.916609 2.059968
exp(wake_fit_1_arima_forecast$upper[7,])
##      80%      95% 
## 10.85018 15.36224
exp(wake_fit_1_arima_forecast$mean[7])-exp(test_df1[7])
## [1] -1.372799
exp(wake_fit_1_arima_forecast$mean[14])
## [1] 5.625412
exp(wake_fit_1_arima_forecast$lower[14,])
##      80%      95% 
## 2.349705 1.480158
exp(wake_fit_1_arima_forecast$upper[14,])
##      80%      95% 
## 13.46776 21.37965
exp(wake_fit_1_arima_forecast$mean[14])-exp(test_df1[14])
## [1] -0.0813081
wake_fit_2_arima  <- Arima(train_df1, order = c(3,1,4))
coeftest(wake_fit_2_arima)
## 
## z test of coefficients:
## 
##     Estimate Std. Error  z value  Pr(>|z|)    
## ar1  1.66735    0.19440   8.5771 < 2.2e-16 ***
## ar2 -1.34881    0.27158  -4.9666 6.814e-07 ***
## ar3  0.63482    0.13938   4.5548 5.244e-06 ***
## ma1 -2.19667    0.19786 -11.1023 < 2.2e-16 ***
## ma2  2.12902    0.38619   5.5129 3.529e-08 ***
## ma3 -1.24457    0.31102  -4.0016 6.291e-05 ***
## ma4  0.35688    0.11466   3.1126  0.001855 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
wake_fit_2_arima_forecast <- forecast::forecast(wake_fit_2_arima ,h=14)
rmse(test_df1,wake_fit_2_arima_forecast$mean) 
## [1] 0.1988205
mae(test_df1,wake_fit_2_arima_forecast$mean) 
## [1] 0.170638
checkresiduals(wake_fit_2_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,4)
## Q* = 6.5951, df = 3, p-value = 0.08599
## 
## Model df: 7.   Total lags used: 10
wake_fit_3_arima <- Arima(train_df1, order = c(3,1,1))
coeftest(wake_fit_3_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value  Pr(>|z|)    
## ar1 -0.053777   0.110265 -0.4877   0.62576    
## ar2 -0.140588   0.062396 -2.2532   0.02425 *  
## ar3 -0.121527   0.054762 -2.2192   0.02647 *  
## ma1 -0.415730   0.103846 -4.0033 6.246e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
wake_fit_3_arima_forecast <- forecast::forecast(wake_fit_3_arima,h=14)
rmse(test_df1,wake_fit_3_arima_forecast$mean) 
## [1] 0.3182116
mae(test_df1,wake_fit_3_arima_forecast$mean) 
## [1] 0.298623
checkresiduals(wake_fit_3_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,1)
## Q* = 23.78, df = 6, p-value = 0.0005732
## 
## Model df: 4.   Total lags used: 10
wake_res_acf <- ggAcf(residuals(wake_fit_1_arima)) + 
  theme_bw(base_size = 15) + ggtitle("")

wake_qqplot <- data.frame(y=residuals(wake_fit_1_arima)) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

png(filename = "arima_res_wake.png", res = 700,
    units = "cm",width = 20, height = 10)
grid.arrange(wake_res_acf,wake_qqplot, ncol=2)
dev.off()
## quartz_off_screen 
##                 2
### Mecklenburg

mecklenburg_reported_cases <- subset(reported_cases,County=="Mecklenburg")
summary(mecklenburg_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1375        Length:1375        Length:1375        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-06-16  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-10-08  
##                                                           Mean   :2021-10-02  
##                                                           3rd Qu.:2022-01-31  
##                                                           Max.   :2022-05-25  
##                                                                               
##    population     new_cases_per_10k
##  Min.   : 68685   Min.   : 0.110   
##  1st Qu.: 68685   1st Qu.: 0.990   
##  Median :120000   Median : 2.040   
##  Mean   :124133   Mean   : 3.982   
##  3rd Qu.:182501   3rd Qu.: 4.330   
##  Max.   :182501   Max.   :47.580   
##                   NA's   :10
mecklenburg_reported_cases$new_cases_per_10k <- 
  LOCF(mecklenburg_reported_cases$new_cases_per_10k)

df_mecklenburg <- mecklenburg_reported_cases %>% 
  group_by(Date) %>% 
  summarise(mean_new_cases = mean(new_cases_per_10k))
df_mecklenburg_1 <- xts(df_mecklenburg$mean_new_cases,order.by = df_mecklenburg$Date)
df_mecklenburg_1 <- df_mecklenburg_1[-c(1,506,507,508),] #making even weekly data
attr(df_mecklenburg_1, 'frequency') <- 7 
periodicity(df_mecklenburg_1)  
## Daily periodicity from 2021-01-04 to 2022-05-22
df_mecklenburg_1_ts <- as.ts(df_mecklenburg_1)
seasonal_decomp_mecklenburg<- decompose(log(df_mecklenburg_1_ts))
deseasonal_decomp_mecklen <- seasadj(seasonal_decomp_mecklenburg)

#Forecasting
train_df_seasonal_mecklen <- ts(log(df_mecklenburg_1_ts)[-c(491:504)], frequency=7)
test_df_seasonal_mecklen <- log(df_mecklenburg_1_ts)[c(491:504)]

train_df1_mecklen <- deseasonal_decomp_mecklen[-c(491:504)]
test_df1_mecklen <- deseasonal_decomp_mecklen[c(491:504)]

lowest_rmse_meck<-Inf
best_mod_meck<-NULL
lowest_mae_meck<-Inf
best_mod_meck_mae<-NULL

for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1_mecklen, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    rmse_mod_1 <- rmse(test_df1_mecklen,forecast_fit$mean)
    if (rmse_mod_1 < lowest_rmse_meck){
      lowest_rmse_meck <- rmse_mod_1 
      best_mod_meck <- arima_mod_1
    }
  }
} #arima(3,1,1) gives the lowest RMSE 

best_mod_meck 
## Series: train_df1_mecklen 
## ARIMA(3,1,1) 
## 
## Coefficients:
##           ar1      ar2      ar3      ma1
##       -0.0472  -0.1851  -0.0397  -0.4183
## s.e.   0.1330   0.0698   0.0635   0.1261
## 
## sigma^2 = 0.1002:  log likelihood = -129.59
## AIC=269.18   AICc=269.3   BIC=290.14
lowest_rmse_meck
## [1] 0.1169349
for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1_mecklen, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    mae_mod <- mae(test_df1_mecklen,forecast_fit$mean)
    if (mae_mod < lowest_mae_meck){
      lowest_mae_meck <- mae_mod
      best_mod_meck_mae <- arima_mod_1
    }
  }
} #arima(3,1,1) gives the lowest mae

best_mod_meck_mae 
## Series: train_df1_mecklen 
## ARIMA(3,1,1) 
## 
## Coefficients:
##           ar1      ar2      ar3      ma1
##       -0.0472  -0.1851  -0.0397  -0.4183
## s.e.   0.1330   0.0698   0.0635   0.1261
## 
## sigma^2 = 0.1002:  log likelihood = -129.59
## AIC=269.18   AICc=269.3   BIC=290.14
lowest_mae_meck
## [1] 0.1015061
meck_mod_1_arima <- Arima(train_df1_mecklen, order = c(4,1,4))
coeftest(meck_mod_1_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value Pr(>|z|)
## ar1  0.179200   2.855669  0.0628   0.9500
## ar2  0.571532   1.826062  0.3130   0.7543
## ar3  0.183666   1.138179  0.1614   0.8718
## ar4 -0.061576   0.292521 -0.2105   0.8333
## ma1 -0.712064   2.857448 -0.2492   0.8032
## ma2 -0.668450   3.339767 -0.2001   0.8414
## ma3  0.228473   0.841678  0.2714   0.7860
## ma4  0.276417   1.369902  0.2018   0.8401
meck_mod_1_arima_forecast <- forecast::forecast(meck_mod_1_arima , h=14)
rmse(test_df1_mecklen,meck_mod_1_arima_forecast$mean) 
## [1] 0.2734803
mae(test_df1_mecklen,meck_mod_1_arima_forecast$mean) 
## [1] 0.2377482
checkresiduals(meck_mod_1_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(4,1,4)
## Q* = 7.3206, df = 3, p-value = 0.06235
## 
## Model df: 8.   Total lags used: 11
exp(meck_mod_1_arima_forecast$mean[1])
## [1] 3.671556
exp(meck_mod_1_arima_forecast$lower[1,])
##      80%      95% 
## 2.478145 2.012571
exp(meck_mod_1_arima_forecast$upper[1,])
##      80%      95% 
## 5.439683 6.698061
exp(meck_mod_1_arima_forecast$mean[1])- exp(test_df1_mecklen[1])
## [1] 0.6454289
exp(meck_mod_1_arima_forecast$mean[7])
## [1] 4.497959
exp(meck_mod_1_arima_forecast$lower[7,])
##      80%      95% 
## 2.606630 1.952778
exp(meck_mod_1_arima_forecast$upper[7,])
##       80%       95% 
##  7.761607 10.360438
exp(meck_mod_1_arima_forecast$mean[7])-exp(test_df1_mecklen[7])
## [1] 0.643501
exp(meck_mod_1_arima_forecast$mean[14])
## [1] 5.36376
exp(meck_mod_1_arima_forecast$lower[14,])
##      80%      95% 
## 2.371565 1.539604
exp(meck_mod_1_arima_forecast$upper[14,])
##      80%      95% 
## 12.13120 18.68658
exp(meck_mod_1_arima_forecast$mean[14])-exp(test_df1_mecklen[14])
## [1] 2.34382
meck_mod_2_arima <- Arima(train_df1_mecklen, order = c(3,1,4))
coeftest(meck_mod_2_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value Pr(>|z|)
## ar1  0.913670   2.674093  0.3417   0.7326
## ar2  0.070303   2.548185  0.0276   0.9780
## ar3 -0.050111   0.183544 -0.2730   0.7848
## ma1 -1.446301   2.676266 -0.5404   0.5889
## ma2  0.221491   3.956001  0.0560   0.9554
## ma3  0.333399   0.923784  0.3609   0.7182
## ma4 -0.043563   0.592916 -0.0735   0.9414
meck_mod_2_arima_forecast <- forecast::forecast(meck_mod_2_arima , h=14)
rmse(test_df1_mecklen,meck_mod_2_arima_forecast$mean) 
## [1] 0.2746343
mae(test_df1_mecklen,meck_mod_2_arima_forecast$mean) 
## [1] 0.2388543
checkresiduals(meck_mod_2_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,4)
## Q* = 3.8301, df = 3, p-value = 0.2804
## 
## Model df: 7.   Total lags used: 10
meck_mod_3_arima <- Arima(train_df1_mecklen, order = c(3,1,1))
coeftest(meck_mod_3_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value  Pr(>|z|)    
## ar1 -0.047185   0.133005 -0.3548 0.7227680    
## ar2 -0.185127   0.069811 -2.6518 0.0080054 ** 
## ar3 -0.039679   0.063536 -0.6245 0.5322946    
## ma1 -0.418331   0.126070 -3.3182 0.0009059 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
meck_mod_3_arima_forecast <- forecast::forecast(meck_mod_3_arima , h=14)
rmse(test_df1_mecklen,meck_mod_3_arima_forecast$mean) 
## [1] 0.1169349
mae(test_df1_mecklen,meck_mod_3_arima_forecast$mean) 
## [1] 0.1015061
checkresiduals(meck_mod_3_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,1)
## Q* = 27.175, df = 6, p-value = 0.0001343
## 
## Model df: 4.   Total lags used: 10
meck_res_acf <- ggAcf(residuals(meck_mod_1_arima)) + 
  theme_bw(base_size = 15) + ggtitle("")

meck_qqplot <- data.frame(y=residuals(meck_mod_1_arima)) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

png(filename = "arima_res_meck.png", res = 700,
    units = "cm",width = 20, height = 10)
grid.arrange(meck_res_acf,meck_qqplot, ncol=2)
dev.off()
## quartz_off_screen 
##                 2
##New Hanover

new_hanover_reported_cases <- subset(reported_cases,County=="New Hanover")
summary(new_hanover_reported_cases)
##     Index               wwtp              County               Date           
##  Length:1011        Length:1011        Length:1011        Min.   :2021-01-03  
##  Class :character   Class :character   Class :character   1st Qu.:2021-05-11  
##  Mode  :character   Mode  :character   Mode  :character   Median :2021-09-15  
##                                                           Mean   :2021-09-14  
##                                                           3rd Qu.:2022-01-19  
##                                                           Max.   :2022-05-25  
##                                                                               
##    population    new_cases_per_10k
##  Min.   :58361   Min.   : 0.300   
##  1st Qu.:58361   1st Qu.: 0.340   
##  Median :58361   Median : 1.710   
##  Mean   :63029   Mean   : 3.529   
##  3rd Qu.:67743   3rd Qu.: 3.600   
##  Max.   :67743   Max.   :48.490   
##                  NA's   :37
new_hanover_reported_cases$new_cases_per_10k <- 
  LOCF(new_hanover_reported_cases$new_cases_per_10k)

df_new_hanover <- new_hanover_reported_cases %>% 
  group_by(Date) %>% 
  summarise(mean_new_cases = mean(new_cases_per_10k))
df_new_hanover_1 <- 
  xts(df_new_hanover$mean_new_cases,order.by = df_new_hanover$Date)
df_new_hanover_1 <- df_new_hanover_1[-c(1,506,507,508),] #making even weekly data
attr(df_new_hanover_1, 'frequency') <- 7 
periodicity(df_new_hanover_1)  
## Daily periodicity from 2021-01-04 to 2022-05-22
df_new_hanover_1_ts <- as.ts(df_new_hanover_1)

new_hanover_seasonal_decomp <- decompose(log(df_new_hanover_1_ts))
new_hanover_deseasonal_decomp <- seasadj(new_hanover_seasonal_decomp)

#forecasting

train_df_seasonal_new_hanover <- log(df_new_hanover_1_ts)[-c(491:504)]
test_df_seasonal_new_hanover <- log(df_new_hanover_1_ts)[c(491:504)]

train_df1_new_hanover<- new_hanover_deseasonal_decomp[-c(491:504)]
test_df1_new_hanover <- new_hanover_deseasonal_decomp[c(491:504)]

lowest_rmse_hanover<-Inf
best_mod_hanover<-NULL

for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1_new_hanover, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    rmse_mod_1 <- rmse(test_df1_new_hanover,forecast_fit$mean)
    if (rmse_mod_1 < lowest_rmse_hanover){
      lowest_rmse_hanover <- rmse_mod_1 
      best_mod_hanover <- arima_mod_1
    }
  }
} 

best_mod_hanover #arima(4,1,4), arima(3,1,3)
## Series: train_df1_new_hanover 
## ARIMA(4,1,4) 
## 
## Coefficients:
##          ar1      ar2     ar3      ar4      ma1     ma2      ma3     ma4
##       0.6073  -0.5023  1.0178  -0.2324  -1.1493  0.6785  -1.2525  0.8249
## s.e.  0.0624   0.0370  0.0346   0.0599   0.0400  0.0242   0.0399  0.0420
## 
## sigma^2 = 0.1368:  log likelihood = -205.8
## AIC=429.6   AICc=429.98   BIC=467.33
lowest_rmse_hanover
## [1] 0.3371953
lowest_mae_hanover<-Inf
best_mod_hanover_mae<-NULL

for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(train_df1_new_hanover, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    mae_mod_1 <- mae(test_df1_new_hanover,forecast_fit$mean)
    if (mae_mod_1 < lowest_mae_hanover){
      lowest_mae_hanover <- mae_mod_1 
      best_mod_hanover_mae <- arima_mod_1
    }
  }
} 

best_mod_hanover_mae
## Series: train_df1_new_hanover 
## ARIMA(4,1,4) 
## 
## Coefficients:
##          ar1      ar2     ar3      ar4      ma1     ma2      ma3     ma4
##       0.6073  -0.5023  1.0178  -0.2324  -1.1493  0.6785  -1.2525  0.8249
## s.e.  0.0624   0.0370  0.0346   0.0599   0.0400  0.0242   0.0399  0.0420
## 
## sigma^2 = 0.1368:  log likelihood = -205.8
## AIC=429.6   AICc=429.98   BIC=467.33
lowest_mae_hanover
## [1] 0.2450774
#since wake, meck have arima(4,1,4) as second best model out of the best models 
#considered, arima(4,1,4) will be used for comparision, allows to accomodate complex data 

new_hanover_mod_1_arima <- Arima(train_df1_new_hanover, order = c(3,1,1))
coeftest(new_hanover_mod_1_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value  Pr(>|z|)    
## ar1 -0.035479   0.100974 -0.3514  0.725309    
## ar2 -0.133723   0.060751 -2.2012  0.027724 *  
## ar3 -0.140904   0.054373 -2.5915  0.009557 ** 
## ma1 -0.463325   0.094406 -4.9078 9.211e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
new_hanover_mod_1_arima_forecast <- forecast::forecast(new_hanover_mod_1_arima , h=14)
rmse(test_df1_new_hanover,new_hanover_mod_1_arima_forecast$mean) 
## [1] 0.3807585
mae(test_df1_new_hanover,new_hanover_mod_1_arima_forecast$mean) 
## [1] 0.3233545
checkresiduals(new_hanover_mod_1_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,1)
## Q* = 30.235, df = 6, p-value = 3.546e-05
## 
## Model df: 4.   Total lags used: 10
new_hanover_mod_2_arima <- Arima(train_df1_new_hanover, order = c(3,1,4))
coeftest(new_hanover_mod_2_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value  Pr(>|z|)    
## ar1  0.430642   0.267663  1.6089    0.1076    
## ar2  0.138449   0.306255  0.4521    0.6512    
## ar3  0.338331   0.450347  0.7513    0.4525    
## ma1 -0.999079   0.241181 -4.1424 3.436e-05 ***
## ma2 -0.033418   0.291343 -0.1147    0.9087    
## ma3 -0.284498   0.632338 -0.4499    0.6528    
## ma4  0.404610   0.311841  1.2975    0.1945    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
new_hanover_mod_2_arima_forecast <- forecast::forecast(new_hanover_mod_2_arima , h=14)
rmse(test_df1_new_hanover,new_hanover_mod_2_arima_forecast$mean) 
## [1] 0.3713594
mae(test_df1_new_hanover,new_hanover_mod_2_arima_forecast$mean) 
## [1] 0.2635143
checkresiduals(new_hanover_mod_2_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,4)
## Q* = 9.1476, df = 3, p-value = 0.02739
## 
## Model df: 7.   Total lags used: 10
new_hanover_mod_3_arima <- Arima(train_df1_new_hanover, order = c(4,1,4))
coeftest(new_hanover_mod_3_arima)
## 
## z test of coefficients:
## 
##      Estimate Std. Error  z value  Pr(>|z|)    
## ar1  0.607271   0.062391   9.7333 < 2.2e-16 ***
## ar2 -0.502302   0.037004 -13.5743 < 2.2e-16 ***
## ar3  1.017773   0.034645  29.3775 < 2.2e-16 ***
## ar4 -0.232440   0.059887  -3.8813 0.0001039 ***
## ma1 -1.149288   0.040039 -28.7041 < 2.2e-16 ***
## ma2  0.678547   0.024190  28.0508 < 2.2e-16 ***
## ma3 -1.252469   0.039911 -31.3814 < 2.2e-16 ***
## ma4  0.824887   0.041974  19.6524 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
new_hanover_mod_3_arima_forecast <- forecast::forecast(new_hanover_mod_3_arima , h=14)
rmse(test_df1_new_hanover,new_hanover_mod_3_arima_forecast$mean) 
## [1] 0.3371953
mae(test_df1_new_hanover,new_hanover_mod_3_arima_forecast$mean) 
## [1] 0.2450774
checkresiduals(new_hanover_mod_3_arima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(4,1,4)
## Q* = 10.834, df = 3, p-value = 0.01266
## 
## Model df: 8.   Total lags used: 11
exp(new_hanover_mod_3_arima_forecast$mean[1])
## [1] 1.428806
exp(new_hanover_mod_3_arima_forecast$lower[1,])
##       80%       95% 
## 0.8892070 0.6917811
exp(new_hanover_mod_3_arima_forecast$upper[1,])
##      80%      95% 
## 2.295852 2.951060
exp(new_hanover_mod_3_arima_forecast$mean[1])-exp(test_df1_new_hanover[1])
## [1] 0.3205985
exp(new_hanover_mod_3_arima_forecast$mean[7])
## [1] 2.041789
exp(new_hanover_mod_3_arima_forecast$lower[7,])
##       80%       95% 
## 1.0914095 0.7834069
exp(new_hanover_mod_3_arima_forecast$upper[7,])
##      80%      95% 
## 3.819742 5.321505
exp(new_hanover_mod_3_arima_forecast$mean[7])-exp(test_df1_new_hanover[7])
## [1] -0.2166181
exp(new_hanover_mod_3_arima_forecast$mean[14])
## [1] 2.291011
exp(new_hanover_mod_3_arima_forecast$lower[14,])
##       80%       95% 
## 0.9472997 0.5935452
exp(new_hanover_mod_3_arima_forecast$upper[14,])
##      80%      95% 
## 5.540727 8.843016
exp(new_hanover_mod_3_arima_forecast$mean[14])-exp(test_df1_new_hanover[14])
## [1] 1.451784
hanover_res_acf <- ggAcf(residuals(new_hanover_mod_3_arima)) + 
  theme_bw(base_size = 15) + ggtitle("")

hanover_qqplot <- data.frame(y=residuals(new_hanover_mod_3_arima)) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

png(filename = "arima_res_hanover.png", res = 700,
    units = "cm",width = 20, height = 10)
grid.arrange(hanover_res_acf,hanover_qqplot, ncol=2)
dev.off()
## quartz_off_screen 
##                 2
#forecasting plots

wake_forecast_plot <- 
  autoplot(wake_fit_1_arima_forecast) + 
  autolayer(wake_fit_1_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1,start = 491), series = "Observed") +
  theme_bw() + ylab("") + 
  ggtitle(NULL) + theme(legend.position = "none") #wake

meck_forecast_plot <- 
  autoplot(meck_mod_1_arima_forecast) + 
  autolayer(meck_mod_1_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1_mecklen,start = 491), series = "Observed") +
  theme_bw() + ylab("")  + 
  ggtitle(NULL) + theme(legend.position = "none") #meck

hanover_forecast_plot <- 
  autoplot(new_hanover_mod_3_arima_forecast) + 
  autolayer(new_hanover_mod_3_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1_new_hanover,start = 491), series = "Observed") +
  theme_bw() + ylab("")+ 
  ggtitle(NULL) + theme(legend.position = "bottom")#new hanover

png(filename = "arima_forecasts.png",units = "cm",
    res = 700, width = 20, height = 15)
grid.arrange(wake_forecast_plot, meck_forecast_plot, 
             hanover_forecast_plot,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1)) 
dev.off()
## quartz_off_screen 
##                 2
#forecasting ARIMA(3,1,1)

wake_forecast_plot_311 <- autoplot(wake_fit_3_arima_forecast) + 
  autolayer(wake_fit_3_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("")+
  ggtitle(NULL) + theme(legend.position = "none") #wake

meck_forecast_plot_311 <- autoplot(meck_mod_3_arima_forecast) + 
  autolayer(meck_mod_3_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1_mecklen,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("")  + xlab("")+
  ggtitle(NULL) + theme(legend.position = "none") #meck

hanover_forecast_plot_311 <- 
  autoplot(new_hanover_mod_1_arima_forecast) + 
  autolayer(new_hanover_mod_1_arima_forecast, series = "Forecasted") +
  autolayer(ts(test_df1_new_hanover,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("")+ 
  ggtitle(NULL) + theme(legend.position = "bottom") #new hanover

SARIMA modelling

#Wake

best_sarima_mod <- NULL
lowest_sarima_mod_rmse <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
    sarima_mod_1 <- Arima(train_df_seasonal, order = c(p,d,q),
                          seasonal = list(order=c(P,D,Q),period=7),
                          method="CSS")
    forecast_fit <- forecast::forecast(sarima_mod_1,14)
    rmse_mod <- rmse(test_df_seasonal,forecast_fit$mean)
    if (rmse_mod < lowest_sarima_mod_rmse){
      lowest_sarima_mod_rmse <- rmse_mod
      best_sarima_mod <- sarima_mod_1
    }
          }
        }
      }
    }
  }
}

best_sarima_mod
## Series: train_df_seasonal 
## ARIMA(2,3,2)(1,1,2)[7] 
## 
## Coefficients:
##           ar1      ar2      ma1     ma2     sar1    sma1     sma2
##       -0.4003  -0.3414  -1.9616  0.9706  -0.9485  0.0086  -0.9719
## s.e.   0.0456   0.0167   0.0109  0.0115   0.0172  0.0168   0.0166
## 
## sigma^2 = 0.1343:  log likelihood = -200.33
lowest_sarima_mod_rmse
## [1] 0.1213027
best_sarima_mod_mae <- NULL
lowest_sarima_mod_mae <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(train_df_seasonal, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            mae_mod <- mae(test_df_seasonal,forecast_fit$mean)
            if (mae_mod < lowest_sarima_mod_mae){
              lowest_sarima_mod_mae <- mae_mod
              best_sarima_mod_mae <- sarima_mod_1
            }
          }
        }
      }
    }
  }
}

best_sarima_mod_mae 
## Series: train_df_seasonal 
## ARIMA(2,3,2)(1,1,2)[7] 
## 
## Coefficients:
##           ar1      ar2      ma1     ma2     sar1    sma1     sma2
##       -0.4003  -0.3414  -1.9616  0.9706  -0.9485  0.0086  -0.9719
## s.e.   0.0456   0.0167   0.0109  0.0115   0.0172  0.0168   0.0166
## 
## sigma^2 = 0.1343:  log likelihood = -200.33
lowest_sarima_mod_mae 
## [1] 0.08304265
wake_sarima_mod_1 <- Arima(train_df_seasonal,
                         order = c(2,3,2), 
                         seasonal =c(1,1,2),method="CSS")
wake_sarima_mod_1_forecast<- forecast::forecast(wake_sarima_mod_1, h=14)
rmse(test_df_seasonal,wake_sarima_mod_1_forecast$mean)
## [1] 0.1213027
mae(test_df_seasonal,wake_sarima_mod_1_forecast$mean)
## [1] 0.08304265
checkresiduals(wake_sarima_mod_1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)(1,1,2)[7]
## Q* = 54.244, df = 7, p-value = 2.105e-09
## 
## Model df: 7.   Total lags used: 14
exp(wake_sarima_mod_1_forecast$mean[1])
## [1] 3.259008
exp(wake_sarima_mod_1_forecast$lower[1,])
##      80%      95% 
## 2.034780 1.585718
exp(wake_sarima_mod_1_forecast$upper[1,])
##      80%      95% 
## 5.219792 6.697996
exp(wake_sarima_mod_1_forecast$mean[1])-exp(test_df_seasonal[1])
## [1] -1.155992
exp(wake_sarima_mod_1_forecast$mean[7])
## [1] 6.321962
exp(wake_sarima_mod_1_forecast$lower[7,])
##      80%      95% 
## 2.480965 1.512080
exp(wake_sarima_mod_1_forecast$upper[7,])
##      80%      95% 
## 16.10954 26.43194
exp(wake_sarima_mod_1_forecast$mean[7])-exp(test_df_seasonal[7])
## [1] 0.7769618
exp(wake_sarima_mod_1_forecast$mean[14])
## [1] 5.723943
exp(wake_sarima_mod_1_forecast$lower[14,])
##       80%       95% 
## 1.0481530 0.4267143
exp(wake_sarima_mod_1_forecast$upper[14,])
##      80%      95% 
## 31.25834 76.78093
exp(wake_sarima_mod_1_forecast$mean[14])-exp(test_df_seasonal[14])
## [1] 1.202276
wake_sarima_mod_2 <- Arima(train_df_seasonal,
                           order = c(2,0,1), 
                           seasonal =c(1,1,2),method="CSS")
wake_sarima_mod_2_forecast<- forecast::forecast(wake_sarima_mod_2, h=14)
rmse(test_df_seasonal,wake_sarima_mod_2_forecast$mean)
## [1] 0.3779045
mae(test_df_seasonal,wake_sarima_mod_2_forecast$mean)
## [1] 0.360243
checkresiduals(wake_sarima_mod_2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,0,1)(1,1,2)[7]
## Q* = 35.492, df = 8, p-value = 2.174e-05
## 
## Model df: 6.   Total lags used: 14
wake_sarima_mod_3 <- Arima(train_df_seasonal,
                           order = c(2,3,2), 
                           seasonal =c(3,1,2),method="CSS")
wake_sarima_mod_3_forecast<- forecast::forecast(wake_sarima_mod_3, h=14)
rmse(test_df_seasonal,wake_sarima_mod_3_forecast$mean)
## [1] 0.5218837
mae(test_df_seasonal,wake_sarima_mod_3_forecast$mean)
## [1] 0.4289307
checkresiduals(wake_sarima_mod_3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)(3,1,2)[7]
## Q* = 54.435, df = 5, p-value = 1.706e-10
## 
## Model df: 9.   Total lags used: 14
wake_res_acf_sarima <- ggAcf(residuals(wake_sarima_mod_1)) + 
  theme_bw(base_size = 15) + ggtitle("")

wake_qqplot_sarima <- data.frame(y=residuals(wake_sarima_mod_1)) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

grid.arrange(wake_res_acf_sarima ,wake_qqplot_sarima, ncol=2)

#Mecklenburg

best_sarima_mod_meck <- NULL
lowest_sarima_mod_meck_rmse <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(train_df_seasonal_mecklen, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            rmse_mod <- rmse(test_df_seasonal_mecklen,forecast_fit$mean)
            if (rmse_mod < lowest_sarima_mod_meck_rmse){
              lowest_sarima_mod_meck_rmse <- rmse_mod
              best_sarima_mod_meck <- sarima_mod_1
            }
          }
        }
      }
    }
  }
}

best_sarima_mod_meck
## Series: train_df_seasonal_mecklen 
## ARIMA(2,0,1)(1,1,2)[7] 
## 
## Coefficients:
##          ar1      ar2      ma1    sar1     sma1    sma2
##       1.0915  -0.1004  -0.6035  0.2582  -1.1374  0.1774
## s.e.  0.0675   0.0666   0.0503  0.1416   0.1340  0.1290
## 
## sigma^2 = 0.1026:  log likelihood = -136.97
lowest_sarima_mod_meck_rmse
## [1] 0.09694679
best_sarima_mod_mae_meck <- NULL
lowest_sarima_mod_meck_mae <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(train_df_seasonal_mecklen, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            mae_mod <- mae(test_df_seasonal_mecklen,forecast_fit$mean)
            if (mae_mod < lowest_sarima_mod_meck_mae){
              lowest_sarima_mod_meck_mae <- mae_mod
              best_sarima_mod_mae_meck <- sarima_mod_1
            }
          }
        }
      }
    }
  }
}

best_sarima_mod_mae_meck
## Series: train_df_seasonal_mecklen 
## ARIMA(2,0,1)(1,1,1)[7] 
## 
## Coefficients:
##          ar1      ar2      ma1    sar1     sma1
##       1.0838  -0.0939  -0.5950  0.0622  -0.9491
## s.e.  0.0672   0.0663   0.0503  0.0536   0.0227
## 
## sigma^2 = 0.1026:  log likelihood = -137.6
lowest_sarima_mod_meck_mae 
## [1] 0.08221429
meck_sarima_mod_1 <- Arima(train_df_seasonal_mecklen,
                         order = c(2,3,2), 
                         seasonal =c(1,1,2),method="CSS")
meck_sarima_mod_1_forecast <- forecast::forecast(meck_sarima_mod_1, h=14)
rmse(test_df_seasonal_mecklen,meck_sarima_mod_1_forecast$mean)
## [1] 0.1487362
mae(test_df_seasonal_mecklen,meck_sarima_mod_1_forecast$mean)
## [1] 0.1118273
checkresiduals(meck_sarima_mod_1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)(1,1,2)[7]
## Q* = 33.844, df = 7, p-value = 1.842e-05
## 
## Model df: 7.   Total lags used: 14
exp(meck_sarima_mod_1_forecast$mean[1])
## [1] 2.552191
exp(meck_sarima_mod_1_forecast$lower[1,])
##      80%      95% 
## 1.670002 1.334165
exp(meck_sarima_mod_1_forecast$upper[1,])
##      80%      95% 
## 3.900402 4.882215
exp(meck_sarima_mod_1_forecast$mean[1])-exp(test_df_seasonal_mecklen[1])
## [1] 0.4455245
exp(meck_sarima_mod_1_forecast$mean[7])
## [1] 2.634972
exp(meck_sarima_mod_1_forecast$lower[7,])
##      80%      95% 
## 1.148288 0.739766
exp(meck_sarima_mod_1_forecast$upper[7,])
##      80%      95% 
## 6.046460 9.385503
exp(meck_sarima_mod_1_forecast$mean[7])-exp(test_df_seasonal_mecklen[7])
## [1] -0.1516949
exp(meck_sarima_mod_1_forecast$mean[14])
## [1] 2.224326
exp(meck_sarima_mod_1_forecast$lower[14,])
##       80%       95% 
## 0.5145110 0.2370394
exp(meck_sarima_mod_1_forecast$upper[14,])
##      80%      95% 
##  9.61617 20.87259
exp(meck_sarima_mod_1_forecast$mean[14])-exp(test_df_seasonal_mecklen[14])
## [1] 0.0409925
meck_sarima_mod_2 <- Arima(train_df_seasonal_mecklen,
                           order = c(2,0,1), 
                           seasonal =c(1,1,2),method="CSS")
meck_sarima_mod_2_forecast <- forecast::forecast(meck_sarima_mod_2, h=14)
rmse(test_df_seasonal_mecklen,meck_sarima_mod_2_forecast$mean)
## [1] 0.09694679
mae(test_df_seasonal_mecklen,meck_sarima_mod_2_forecast$mean)
## [1] 0.08353163
checkresiduals(meck_sarima_mod_2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,0,1)(1,1,2)[7]
## Q* = 50.493, df = 8, p-value = 3.286e-08
## 
## Model df: 6.   Total lags used: 14
meck_sarima_mod_3 <- Arima(train_df_seasonal_mecklen,
                           order = c(2,3,2), 
                           seasonal =c(3,1,2),method="CSS")
meck_sarima_mod_3_forecast <- forecast::forecast(meck_sarima_mod_3, h=14)
rmse(test_df_seasonal_mecklen,meck_sarima_mod_3_forecast$mean)
## [1] 0.215006
mae(test_df_seasonal_mecklen,meck_sarima_mod_3_forecast$mean)
## [1] 0.181207
checkresiduals(meck_sarima_mod_3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)(3,1,2)[7]
## Q* = 40.792, df = 5, p-value = 1.033e-07
## 
## Model df: 9.   Total lags used: 14
meck_res_acf_sarima <- ggAcf(residuals(meck_sarima_mod_1)) + 
  theme_bw(base_size = 15) + ggtitle("")

meck_qqplot_sarima <- data.frame(y=residuals(meck_sarima_mod_1 )) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

grid.arrange(meck_res_acf_sarima ,meck_qqplot_sarima, ncol=2)

#New Hanover

best_sarima_mod_hanover <- NULL
lowest_sarima_mod_hanover_rmse <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(train_df_seasonal_new_hanover, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            rmse_mod <- rmse(test_df_seasonal_new_hanover,forecast_fit$mean)
            if (rmse_mod < lowest_sarima_mod_hanover_rmse){
              lowest_sarima_mod_hanover_rmse <- rmse_mod
              best_sarima_mod_hanover <- sarima_mod_1
            }
          }
        }
      }
    }
  }
}

best_sarima_mod_hanover
## Series: train_df_seasonal_new_hanover 
## ARIMA(3,3,3)(2,1,3)[7] 
## 
## Coefficients:
##           ar1      ar2      ar3      ma1      ma2     ma3     sar1     sar2
##       -1.4066  -0.6461  -0.2114  -0.8613  -0.8627  0.8074  -0.9656  -0.5930
## s.e.   0.0562   0.0808   0.0514   0.0373   0.0575  0.0369   0.1182   0.1257
##         sma1     sma2     sma3
##       0.1722  -0.1534  -0.4581
## s.e.  0.1238   0.1024   0.0990
## 
## sigma^2 = 0.2335:  log likelihood = -335.12
lowest_sarima_mod_hanover_rmse
## [1] 0.2774404
best_sarima_mod_mae_hanover <- NULL
lowest_sarima_mod_hanover_mae <- Inf

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(train_df_seasonal_new_hanover, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            mae_mod <- mae(test_df_seasonal_new_hanover,forecast_fit$mean)
            if (mae_mod < lowest_sarima_mod_hanover_mae){
              lowest_sarima_mod_hanover_mae <- mae_mod
              best_sarima_mod_mae_hanover <- sarima_mod_1
            }
          }
        }
      }
    }
  }
}

best_sarima_mod_mae_hanover
## Series: train_df_seasonal_new_hanover 
## ARIMA(2,3,2)(3,1,2)[7] 
## 
## Coefficients:
##          ar1      ar2      ma1     ma2     sar1    sar2    sar3     sma1
##       -0.494  -0.1715  -1.9631  0.9687  -0.7427  0.2262  0.1693  -0.1067
## s.e.   0.042   0.0455   0.0097  0.0102   0.0834  0.0608  0.0426   0.0863
##          sma2
##       -0.7766
## s.e.   0.0835
## 
## sigma^2 = 0.175:  log likelihood = -270.02
lowest_sarima_mod_hanover_mae 
## [1] 0.2228928
hanover_sarima_mod_1 <- Arima(train_df_seasonal_new_hanover,
                           order = c(2,3,2), 
                           seasonal =c(1,1,2),method="CSS")
hanover_sarima_mod_1_forecast <- forecast::forecast(hanover_sarima_mod_1, h=14)
rmse(test_df_seasonal_new_hanover,hanover_sarima_mod_1_forecast$mean)
## [1] 0.500865
mae(test_df_seasonal_new_hanover,hanover_sarima_mod_1_forecast$mean)
## [1] 0.3788101
checkresiduals(hanover_sarima_mod_1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)
## Q* = 152.08, df = 6, p-value < 2.2e-16
## 
## Model df: 4.   Total lags used: 10
exp(hanover_sarima_mod_1_forecast$mean[1])
## [1] 1.417873
exp(hanover_sarima_mod_1_forecast$lower[1,])
##       80%       95% 
## 0.7340845 0.5180890
exp(hanover_sarima_mod_1_forecast$upper[1,])
##      80%      95% 
## 2.738599 3.880344
exp(hanover_sarima_mod_1_forecast$mean[1])-exp(test_df1_new_hanover[1])
## [1] 0.309665
exp(hanover_sarima_mod_1_forecast$mean[7])
## [1] 1.869173
exp(hanover_sarima_mod_1_forecast$lower[7,])
##       80%       95% 
## 0.3548533 0.1472504
exp(hanover_sarima_mod_1_forecast$upper[7,])
##       80%       95% 
##  9.845783 23.726986
exp(hanover_sarima_mod_1_forecast$mean[7])-exp(test_df1_new_hanover[7])
## [1] -0.3892341
exp(hanover_sarima_mod_1_forecast$mean[14])
## [1] 2.142046
exp(hanover_sarima_mod_1_forecast$lower[14,])
##        80%        95% 
## 0.13604701 0.03161997
exp(hanover_sarima_mod_1_forecast$upper[14,])
##       80%       95% 
##  33.72628 145.10954
exp(hanover_sarima_mod_1_forecast$mean[14])-exp(test_df1_new_hanover[14])
## [1] 1.302819
hanover_sarima_mod_2 <- Arima(train_df_seasonal_new_hanover,
                           order = c(2,0,1), 
                           seasonal =c(1,1,2),method="CSS")
hanover_sarima_mod_2_forecast <- forecast::forecast(hanover_sarima_mod_2, h=14)
rmse(test_df_seasonal_new_hanover,hanover_sarima_mod_2_forecast$mean)
## [1] 0.5744074
mae(test_df_seasonal_new_hanover,hanover_sarima_mod_2_forecast$mean)
## [1] 0.5232856
checkresiduals(hanover_sarima_mod_2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,0,1) with non-zero mean
## Q* = 110.61, df = 6, p-value < 2.2e-16
## 
## Model df: 4.   Total lags used: 10
hanover_sarima_mod_3 <- Arima(train_df_seasonal_new_hanover,
                              order = c(2,3,2), 
                              seasonal =c(3,1,2),method="CSS")
hanover_sarima_mod_3_forecast <- forecast::forecast(hanover_sarima_mod_3, h=14)
rmse(test_df_seasonal_new_hanover,hanover_sarima_mod_3_forecast$mean)
## [1] 0.500865
mae(test_df_seasonal_new_hanover,hanover_sarima_mod_3_forecast$mean)
## [1] 0.3788101
checkresiduals(hanover_sarima_mod_3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,3,2)
## Q* = 152.08, df = 6, p-value < 2.2e-16
## 
## Model df: 4.   Total lags used: 10
hanover_res_acf_sarima <- ggAcf(residuals(hanover_sarima_mod_1)) + 
  theme_bw(base_size = 15) + ggtitle("")

hanover_qqplot_sarima <- data.frame(y=residuals(hanover_sarima_mod_1)) %>% 
  ggplot(aes(sample=y)) + geom_qq() + geom_qq_line() + 
  theme_bw(base_size = 15) + ylab("Sample Quantiles") + xlab("Theoretical Quantiles")

grid.arrange(hanover_res_acf_sarima ,hanover_qqplot_sarima, ncol=2)

#Forecasting plots

wake_forecast_plot_sarima <- autoplot(wake_sarima_mod_1_forecast) + 
  autolayer(wake_sarima_mod_1_forecast, series = "Forecasted") +
  autolayer(ts(test_df_seasonal,start = 71,frequency = 7), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") +
  ggtitle(NULL) + theme(legend.position = "none") #wake

meck_forecast_plot_sarima <- autoplot(meck_sarima_mod_1_forecast) + 
  autolayer(meck_sarima_mod_1_forecast, series = "Forecasted") +
  autolayer(ts(test_df_seasonal_mecklen,start = 71,frequency = 7), series = "Observed") +
  theme_bw(base_size = 15) + ylab("")  + 
  ggtitle(NULL) + theme(legend.position = "none") #meck

hanover_forecast_plot_sarima <- 
  autoplot(hanover_sarima_mod_1_forecast) + 
  autolayer(hanover_sarima_mod_1_forecast, series = "Forecasted") +
  autolayer(ts(test_df_seasonal_new_hanover,start = 71,frequency = 7), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") +
  ggtitle(NULL) + theme(legend.position = "bottom") #new hanover

png(filename = "sarima_forecasts.png",units = "cm",
    res = 700, width = 20, height = 15)
grid.arrange(wake_forecast_plot_sarima, meck_forecast_plot_sarima, 
             hanover_forecast_plot_sarima,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1)) 
dev.off()
## quartz_off_screen 
##                 2

Incorporating wastewater and weather information

ARIMAX modelling- wastewater information only

#wake#

cases <- xts(full_cases_wastewater_weather_data$mean_new_cases,
    order.by = full_cases_wastewater_data$Date)
attr(cases , 'frequency') <- 7 
cases <- cases[-c(505,506,507)]
cases <- as.ts(cases)
cases_decompose <- decompose(log(cases))
cases_deseasonalise <- seasadj(cases_decompose)

viral_gene <- xts(full_cases_wastewater_weather_data$full_viral_gene_copies_per_person,
                  order.by = full_cases_wastewater_data$Date)
attr(viral_gene , 'frequency') <- 7 
viral_gene  <- viral_gene[-c(505,506,507)]
viral_gene  <- as.ts(viral_gene)
viral_decompose <- decompose(log(viral_gene))
viral_deseasonalise <- seasadj(viral_decompose)

cases_train <- cases_deseasonalise[-c(491:504)]
cases_test <- cases_deseasonalise[c(491:504)]

viral_train <- viral_deseasonalise[-c(491:504)]
viral_test <- viral_deseasonalise[c(491:504)]

wastewater_mod_1_wake <- Arima(cases_train ,order = c(3,1,1),
                                xreg = viral_train)
coeftest(wastewater_mod_1_wake) # weakly insignificant
## 
## z test of coefficients:
## 
##       Estimate Std. Error z value  Pr(>|z|)    
## ar1  -0.047442   0.109886 -0.4317   0.66593    
## ar2  -0.139566   0.062292 -2.2405   0.02506 *  
## ar3  -0.120544   0.054927 -2.1946   0.02819 *  
## ma1  -0.424449   0.103710 -4.0926 4.265e-05 ***
## xreg  0.016504   0.023024  0.7168   0.47349    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_wake <- forecast::forecast(wastewater_mod_1_wake, h=14,
                           xreg = viral_test)
rmse(cases_test,forecast_mod_1_wake$mean) #0.31
## [1] 0.31084
mae(cases_test,forecast_mod_1_wake$mean) #0.29
## [1] 0.2905952
checkresiduals(wastewater_mod_1_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 22.404, df = 5, p-value = 0.0004386
## 
## Model df: 5.   Total lags used: 10
wastewater_mod_2_wake <- Arima(cases_train ,order = c(4,1,4),
                               xreg = viral_train)
coeftest(wastewater_mod_2_wake) # weakly insignificant
## 
## z test of coefficients:
## 
##       Estimate Std. Error z value Pr(>|z|)   
## ar1  -0.214677   0.362355 -0.5924 0.553549   
## ar2   0.273832   0.122021  2.2441 0.024824 * 
## ar3  -0.551651   0.213174 -2.5878 0.009659 **
## ar4  -0.158971   0.139375 -1.1406 0.254038   
## ma1  -0.266203   0.363780 -0.7318 0.464311   
## ma2  -0.501328   0.184487 -2.7174 0.006579 **
## ma3   0.640330   0.274584  2.3320 0.019701 * 
## ma4  -0.053557   0.232556 -0.2303 0.817861   
## xreg  0.011573   0.022317  0.5186 0.604072   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_2_wake <- forecast::forecast(wastewater_mod_2_wake, h=14,
                           xreg = viral_test)
rmse(cases_test,forecast_mod_2_wake$mean)
## [1] 0.2878107
mae(cases_test,forecast_mod_2_wake$mean) 
## [1] 0.2638574
checkresiduals(wastewater_mod_2_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 13.8, df = 3, p-value = 0.00319
## 
## Model df: 9.   Total lags used: 12
exp(forecast_mod_2_wake$mean[1])
## [1] 5.383733
exp(forecast_mod_2_wake$lower[1,])
##      80%      95% 
## 3.497436 2.783436
exp(forecast_mod_2_wake$upper[1,])
##       80%       95% 
##  8.287381 10.413238
exp(forecast_mod_2_wake$mean[1]) - exp(cases_test[1])
## [1] -3.333811
exp(forecast_mod_2_wake$mean[7])
## [1] 5.644255
exp(forecast_mod_2_wake$lower[7,])
##      80%      95% 
## 2.929634 2.070394
exp(forecast_mod_2_wake$upper[7,])
##      80%      95% 
## 10.87426 15.38722
exp(forecast_mod_2_wake$mean[7])-exp(cases_test[7])
## [1] -1.353998
exp(forecast_mod_2_wake$mean[14])
## [1] 5.6726
exp(forecast_mod_2_wake$lower[14,])
##      80%      95% 
## 2.375574 1.498512
exp(forecast_mod_2_wake$upper[14,])
##      80%      95% 
## 13.54552 21.47357
exp(forecast_mod_2_wake$mean[14])-exp(cases_test[14])
## [1] -0.03411993
#Mecklenburg#

glimpse(full_cases_wastewater_weather_data_meck)
## Rows: 507
## Columns: 6
## $ Date                              <date> 2021-01-04, 2021-01-05, 2021-01-06,…
## $ mean_new_cases                    <dbl> 5.550, 11.860, 8.345, 7.765, 8.730, …
## $ mean_viral_gene_copies_per_person <dbl> 48863073, 16458455, 28467455, NA, NA…
## $ full_viral_gene_copies_per_person <dbl> 48863073, 16458455, 28467455, 284674…
## $ mean_precipation                  <dbl> 0.0023076923, 0.0123076923, 0.019166…
## $ TAVG                              <int> 47, 45, 42, 39, 38, 39, 37, 39, 45, …
cases_meck <- xts(full_cases_wastewater_weather_data_meck$mean_new_cases,
             order.by = full_cases_wastewater_data_meck$Date)
attr(cases_meck , 'frequency') <- 7 
cases_meck <- cases_meck[-c(505,506,507)]
cases_meck <- as.ts(cases_meck)
cases_decompose_meck <- decompose(log(cases_meck))
cases_deseasonalise_meck <- seasadj(cases_decompose_meck)

viral_gene_meck <- xts(full_cases_wastewater_weather_data_meck$full_viral_gene_copies_per_person,
                  order.by = full_cases_wastewater_data_meck$Date)
attr(viral_gene_meck , 'frequency') <- 7 
viral_gene_meck  <- viral_gene_meck[-c(505,506,507)]
viral_gene_meck  <- as.ts(viral_gene_meck)
viral_decompose_meck <- decompose(log(viral_gene_meck))
viral_deseasonalise_meck <- seasadj(viral_decompose_meck)

cases_train_meck <- cases_deseasonalise_meck[-c(491:504)]
cases_test_meck <- cases_deseasonalise_meck[c(491:504)]

viral_train_meck <- viral_deseasonalise_meck[-c(491:504)]
viral_test_meck <- viral_deseasonalise_meck[c(491:504)]

wastewater_mod_1_meck <- Arima(cases_train_meck,order = c(3,1,1),
                               xreg = viral_train_meck)
coeftest(wastewater_mod_1_meck ) #wastewater insignificant
## 
## z test of coefficients:
## 
##       Estimate Std. Error z value Pr(>|z|)   
## ar1  -0.057166   0.140431 -0.4071 0.683952   
## ar2  -0.186898   0.071762 -2.6044 0.009203 **
## ar3  -0.036454   0.064648 -0.5639 0.572831   
## ma1  -0.400150   0.134646 -2.9719 0.002960 **
## xreg -0.032820   0.031448 -1.0436 0.296662   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_meck <- forecast::forecast(wastewater_mod_1_meck, h=14,
                           xreg = viral_test_meck)
rmse(cases_test_meck,forecast_mod_1_meck$mean) 
## [1] 0.113133
mae(cases_test_meck,forecast_mod_1_meck$mean) 
## [1] 0.09810705
checkresiduals(wastewater_mod_1_meck) #wastewater improves forecast

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 29.438, df = 5, p-value = 1.902e-05
## 
## Model df: 5.   Total lags used: 10
wastewater_mod_2_meck <- Arima(cases_train_meck ,order = c(4,1,4),
                               xreg = viral_train_meck, method = "CSS")
coeftest(wastewater_mod_2_meck) #insignificant
## 
## z test of coefficients:
## 
##       Estimate Std. Error z value Pr(>|z|)
## ar1   0.171347   1.706150  0.1004   0.9200
## ar2   0.391647   2.200231  0.1780   0.8587
## ar3   0.434785   0.919334  0.4729   0.6363
## ar4  -0.134944   0.350314 -0.3852   0.7001
## ma1  -0.702922   1.712747 -0.4104   0.6815
## ma2  -0.478628   3.173802 -0.1508   0.8801
## ma3  -0.118355   1.840228 -0.0643   0.9487
## ma4   0.439516   0.336950  1.3044   0.1921
## xreg -0.036932   0.027152 -1.3602   0.1738
forecast_mod_2_meck <- forecast::forecast(wastewater_mod_2_meck, h=14,
                           xreg = viral_test_meck)
rmse(cases_test_meck,forecast_mod_2_meck$mean) 
## [1] 0.293157
mae(cases_test_meck,forecast_mod_2_meck$mean) 
## [1] 0.2584049
checkresiduals(wastewater_mod_2_meck) #wastewater does not improve forecast

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 8.8691, df = 3, p-value = 0.03108
## 
## Model df: 9.   Total lags used: 12
exp(forecast_mod_2_meck$mean[1])
## [1] 3.715387
exp(forecast_mod_2_meck$lower[1,])
##      80%      95% 
## 2.510192 2.039657
exp(forecast_mod_2_meck$upper[1,])
##      80%      95% 
## 5.499219 6.767852
exp(forecast_mod_2_meck$mean[1])-exp(cases_test_meck[1])
## [1] 0.6892594
exp(forecast_mod_2_meck$mean[7])
## [1] 4.55529
exp(forecast_mod_2_meck$lower[7,])
##      80%      95% 
## 2.619779 1.954714
exp(forecast_mod_2_meck$upper[7,])
##      80%      95% 
##  7.92077 10.61570
exp(forecast_mod_2_meck$mean[7])-exp(cases_test_meck[7])
## [1] 0.7008319
exp(forecast_mod_2_meck$mean[14])
## [1] 5.46859
exp(forecast_mod_2_meck$lower[14,])
##      80%      95% 
## 2.365444 1.517898
exp(forecast_mod_2_meck$upper[14,])
##      80%      95% 
## 12.64265 19.70190
exp(forecast_mod_2_meck$mean[14])-exp(cases_test_meck[14])
## [1] 2.448649
#new hanover#

glimpse(full_cases_wastewater_weather_data_hanover)
## Rows: 507
## Columns: 5
## $ Date                              <date> 2021-01-04, 2021-01-05, 2021-01-06,…
## $ mean_new_cases                    <dbl> 3.770, 9.420, 8.220, 7.200, 5.765, 5…
## $ full_viral_gene_copies_per_person <dbl> 5659256, 5659256, 5659256, 5659256, …
## $ mean_precipation                  <dbl> 0.012307692, 0.017142857, 0.08400000…
## $ mean_temp                         <dbl> 49.33333, 42.66667, 42.66667, 41.000…
cases_hanover <- xts(full_cases_wastewater_weather_data_hanover$mean_new_cases,
                  order.by = full_cases_wastewater_data_hanover$Date)
attr(cases_hanover , 'frequency') <- 7 
cases_hanover <- cases_hanover[-c(505,506,507)]
cases_hanover <- as.ts(cases_hanover)
cases_decompose_hanover <- decompose(log(cases_hanover))
cases_deseasonalise_hanover <- seasadj(cases_decompose_hanover)

viral_gene_hanover <- xts(full_cases_wastewater_weather_data_hanover$full_viral_gene_copies_per_person,
                       order.by = full_cases_wastewater_data_hanover$Date)
attr(viral_gene_hanover , 'frequency') <- 7 
viral_gene_hanover  <- viral_gene_hanover[-c(505,506,507)]
viral_gene_hanover  <- as.ts(viral_gene_hanover)
viral_decompose_hanover <- decompose(log(viral_gene_hanover))
viral_deseasonalise_hanover <- seasadj(viral_decompose_hanover)

cases_train_hanover <- cases_deseasonalise_hanover[-c(491:504)]
cases_test_hanover <- cases_deseasonalise_hanover[c(491:504)]

viral_train_hanover <- viral_deseasonalise_hanover[-c(491:504)]
viral_test_hanover <- viral_deseasonalise_hanover[c(491:504)]

wastewater_mod_1_hanover <- Arima(cases_train_hanover ,order = c(3,1,1),
                                  xreg = viral_train_hanover)
coeftest(wastewater_mod_1_hanover) #wastewater insignificant
## 
## z test of coefficients:
## 
##        Estimate Std. Error z value  Pr(>|z|)    
## ar1   0.0021883  0.0926492  0.0236   0.98116    
## ar2  -0.1218986  0.0578334 -2.1078   0.03505 *  
## ar3  -0.1434527  0.0531366 -2.6997   0.00694 ** 
## ma1  -0.5062411  0.0847791 -5.9713 2.354e-09 ***
## xreg  0.0659619  0.0263188  2.5063   0.01220 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_hanover <- forecast::forecast(wastewater_mod_1_hanover, h=14,
                                             xreg = viral_test_hanover)
rmse(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.3605596
mae(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.295442
checkresiduals(wastewater_mod_1_hanover) #not improve forecast

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 31.446, df = 5, p-value = 7.646e-06
## 
## Model df: 5.   Total lags used: 10
wastewater_mod_1_hanover <- Arima(cases_train_hanover ,order = c(4,1,4),
                               xreg = viral_train_hanover)
coeftest(wastewater_mod_1_hanover) #wastewater insignificant
## 
## z test of coefficients:
## 
##       Estimate Std. Error  z value  Pr(>|z|)    
## ar1   0.602240   0.063301   9.5139 < 2.2e-16 ***
## ar2  -0.500627   0.038988 -12.8405 < 2.2e-16 ***
## ar3   1.014573   0.035938  28.2312 < 2.2e-16 ***
## ar4  -0.226315   0.059438  -3.8076 0.0001403 ***
## ma1  -1.145547   0.040934 -27.9850 < 2.2e-16 ***
## ma2   0.676182   0.026457  25.5582 < 2.2e-16 ***
## ma3  -1.249619   0.031691 -39.4310 < 2.2e-16 ***
## ma4   0.817413   0.039434  20.7286 < 2.2e-16 ***
## xreg  0.033476   0.023612   1.4178 0.1562578    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_hanover <- forecast::forecast(wastewater_mod_1_hanover, h=14,
                                xreg = viral_test_hanover)
rmse(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.3494999
mae(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.2473775
checkresiduals(wastewater_mod_1_hanover) #not improve forecast

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 17.245, df = 3, p-value = 0.0006294
## 
## Model df: 9.   Total lags used: 12
#arimax(4,1,4) plots with only wastewater information#

arimax_414_plot_wake <- autoplot(forecast_mod_2_wake) + 
  autolayer(forecast_mod_2_wake, series = "Forecasted") +
  autolayer(ts(cases_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time") +
  ggtitle(NULL) + theme(legend.position = "none") #wake

arimax_414_plot_meck <- autoplot(forecast_mod_2_meck) + 
  autolayer(forecast_mod_2_meck, series = "Forecasted") +
  autolayer(ts(cases_test_meck,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time") +
  ggtitle(NULL) + theme(legend.position = "none") 

arimax_414_plot_hanover <- autoplot(forecast_mod_1_hanover) + 
  autolayer(forecast_mod_1_hanover, series = "Forecasted") +
  autolayer(ts(cases_test_hanover,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time") +
  ggtitle(NULL) + theme(legend.position = "bottom") 

png(filename = "arimax414_plots.png", res = 700, units = "cm",
    width = 20, height = 18)
grid.arrange(arimax_414_plot_wake,
             arimax_414_plot_meck,
             arimax_414_plot_hanover,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2

ARIMAX modelling - wastewater and weather information

#wake#

precipation_wake <- xts(full_cases_wastewater_weather_data$mean_precipation,
                        order.by = full_cases_wastewater_weather_data$Date)
attr(precipation_wake,'frequency') <- 7
precipation_wake <- precipation_wake[-c(505,506,507)]
precipation_wake <- as.ts(precipation_wake)

temp_wake <- xts(full_cases_wastewater_weather_data$TAVG,
                 order.by = full_cases_wastewater_weather_data$Date)
attr(temp_wake,'frequency') <- 7
temp_wake <- temp_wake[-c(505,506,507)]
temp_wake <- as.ts(temp_wake)

precipation_wake_train <- ts(precipation_wake[-c(491:504)])
precipation_wake_test <- ts(precipation_wake[c(491:504)])

temp_wake_train <- ts(temp_wake[-c(491:504)])
temp_wake_test <- ts(temp_wake[c(491:504)])

vars_wake <- ts.union(viral_train,precipation_wake_train,
                 temp_wake_train)

vars_test_wake <- ts.union(viral_test,precipation_wake_test,
                      temp_wake_test)

wastewater_weather_mod_1_wake <- Arima(cases_train ,order = c(3,1,1),
                               xreg = vars_wake)
coeftest(wastewater_weather_mod_1_wake) # weakly insignificant
## 
## z test of coefficients:
## 
##                           Estimate  Std. Error z value  Pr(>|z|)    
## ar1                    -0.04725429  0.10876295 -0.4345   0.66395    
## ar2                    -0.13669089  0.06187876 -2.2090   0.02717 *  
## ar3                    -0.12606682  0.05473052 -2.3034   0.02126 *  
## ma1                    -0.42295848  0.10276067 -4.1160 3.856e-05 ***
## viral_train             0.01789114  0.02310192  0.7744   0.43867    
## precipation_wake_train -0.04699524  0.04599086 -1.0218   0.30686    
## temp_wake_train        -0.00039493  0.00237290 -0.1664   0.86782    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_wake <- forecast::forecast(wastewater_weather_mod_1_wake, h=14,
                           xreg = vars_test_wake)
rmse(cases_test,forecast_mod_1_wake$mean) 
## [1] 0.3054778
mae(cases_test,forecast_mod_1_wake$mean) 
## [1] 0.2848204
checkresiduals(wastewater_weather_mod_1_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 22.532, df = 3, p-value = 5.054e-05
## 
## Model df: 7.   Total lags used: 10
wastewater_weather_mod_2_wake <- Arima(cases_train ,order = c(4,1,4),
                               xreg = vars_wake)
coeftest(wastewater_weather_mod_2_wake) # insignificant
## 
## z test of coefficients:
## 
##                          Estimate Std. Error  z value  Pr(>|z|)    
## ar1                    -0.2758694  0.0662381  -4.1648 3.116e-05 ***
## ar2                     0.6285461  0.0467118  13.4558 < 2.2e-16 ***
## ar3                     0.7855470  0.0401564  19.5622 < 2.2e-16 ***
## ar4                    -0.3033917  0.0630207  -4.8142 1.478e-06 ***
## ma1                    -0.2314046  0.0446697  -5.1803 2.215e-07 ***
## ma2                    -0.8874472  0.0303010 -29.2878 < 2.2e-16 ***
## ma3                    -0.5332980  0.0277722 -19.2026 < 2.2e-16 ***
## ma4                     0.8024096  0.0411428  19.5030 < 2.2e-16 ***
## viral_train            -0.0019084  0.0216879  -0.0880    0.9299    
## precipation_wake_train -0.0397074  0.0447221  -0.8879    0.3746    
## temp_wake_train         0.0010651  0.0022755   0.4681    0.6397    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_2_wake <- forecast::forecast(wastewater_weather_mod_2_wake, h=14,
                           xreg = vars_test_wake)
rmse(cases_test,forecast_mod_2_wake$mean) 
## [1] 0.2082233
mae(cases_test,forecast_mod_2_wake$mean)
## [1] 0.1842068
checkresiduals(wastewater_weather_mod_2_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 14.074, df = 3, p-value = 0.002806
## 
## Model df: 11.   Total lags used: 14
exp(forecast_mod_2_wake$mean[1])
## [1] 5.554114
exp(forecast_mod_2_wake$lower[1,])
##     80%     95% 
## 3.62983 2.89799
exp(forecast_mod_2_wake$upper[1,])
##       80%       95% 
##  8.498521 10.644682
exp(forecast_mod_2_wake$mean[1])-exp(cases_test[1])
## [1] -3.16343
exp(forecast_mod_2_wake$mean[7])
## [1] 6.454832
exp(forecast_mod_2_wake$lower[7,])
##      80%      95% 
## 3.547241 2.583795
exp(forecast_mod_2_wake$upper[7,])
##      80%      95% 
## 11.74571 16.12545
exp(forecast_mod_2_wake$mean[7])-exp(cases_test[7])
## [1] -0.5434207
exp(forecast_mod_2_wake$mean[14])
## [1] 7.376171
exp(forecast_mod_2_wake$lower[14,])
##      80%      95% 
## 3.111867 1.970646
exp(forecast_mod_2_wake$upper[14,])
##      80%      95% 
## 17.48401 27.60917
exp(forecast_mod_2_wake$mean[14]) -exp(cases_test[14])
## [1] 1.669451
wake_forecast_arimax_weather_plot <- autoplot(forecast_mod_1_wake) + 
  autolayer(forecast_mod_1_wake, series = "Forecasted") +
  autolayer(ts(cases_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + 
  xlab("") + ggtitle(NULL) + theme(legend.position = "none") 

#mecklenburg#

precipation_meck <- xts(full_cases_wastewater_weather_data_meck$mean_precipation,
                        order.by = full_cases_wastewater_weather_data_meck$Date)
attr(precipation_meck,'frequency') <- 7
precipation_meck <- precipation_meck[-c(505,506,507)]
precipation_meck <- as.ts(precipation_meck)

temp_meck <- xts(full_cases_wastewater_weather_data_meck$TAVG,
                 order.by = full_cases_wastewater_weather_data_meck$Date)
attr(temp_meck,'frequency') <- 7
temp_meck <- temp_meck[-c(505,506,507)]
temp_meck <- as.ts(temp_meck)

precipation_meck_train <- ts(precipation_meck[-c(491:504)])
precipation_meck_test <- ts(precipation_meck[c(491:504)])

temp_meck_train <- ts(temp_meck[-c(491:504)])
temp_meck_test <- ts(temp_meck[c(491:504)])

vars_meck <- ts.union(viral_train_meck,precipation_meck_train,
                 temp_meck_train)

vars_test_meck <- ts.union(viral_test_meck,precipation_meck_test,
                      temp_meck_test)

wastewater_weather_mod_1_meck <- Arima(cases_train_meck ,order = c(3,1,1),
                                       xreg = vars_meck)
coeftest(wastewater_weather_mod_1_meck) # weakly insignificant
## 
## z test of coefficients:
## 
##                          Estimate Std. Error z value Pr(>|z|)   
## ar1                    -0.0616022  0.1398544 -0.4405 0.659594   
## ar2                    -0.1854175  0.0721950 -2.5683 0.010220 * 
## ar3                    -0.0361904  0.0648499 -0.5581 0.576801   
## ma1                    -0.4016311  0.1339991 -2.9973 0.002724 **
## viral_train_meck       -0.0268804  0.0316277 -0.8499 0.395381   
## precipation_meck_train -0.0492122  0.0526277 -0.9351 0.349737   
## temp_meck_train         0.0029893  0.0025311  1.1810 0.237591   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_meck <- forecast::forecast(wastewater_weather_mod_1_meck, h=14,
                           xreg = vars_test_meck)
rmse(cases_test_meck,forecast_mod_1_meck$mean) 
## [1] 0.1111443
mae(cases_test_meck,forecast_mod_1_meck$mean) 
## [1] 0.09782189
checkresiduals(wastewater_weather_mod_1_meck)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 30.7, df = 3, p-value = 9.832e-07
## 
## Model df: 7.   Total lags used: 10
wastewater_weather_mod_2_meck <- Arima(cases_train_meck ,order = c(4,1,4),
                                       xreg = vars_meck, method = "CSS")
coeftest(wastewater_weather_mod_2_meck) # insignificant
## 
## z test of coefficients:
## 
##                          Estimate Std. Error z value Pr(>|z|)   
## ar1                     0.2760518  0.2869270  0.9621  0.33600   
## ar2                     0.1826836  0.3856264  0.4737  0.63569   
## ar3                     0.5869433  0.3106909  1.8892  0.05887 . 
## ar4                    -0.1862630  0.1151955 -1.6169  0.10589   
## ma1                    -0.8198434  0.2867637 -2.8590  0.00425 **
## ma2                    -0.1946741  0.5414328 -0.3596  0.71918   
## ma3                    -0.3750422  0.4990118 -0.7516  0.45231   
## ma4                     0.5311884  0.2227153  2.3851  0.01708 * 
## viral_train_meck       -0.0308471  0.0268236 -1.1500  0.25014   
## precipation_meck_train -0.0502851  0.0506466 -0.9929  0.32078   
## temp_meck_train         0.0037193  0.0022594  1.6461  0.09974 . 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_2_meck <- forecast::forecast(wastewater_weather_mod_2_meck, h=14,
                           xreg = vars_test_meck)
rmse(cases_test_meck,forecast_mod_2_meck$mean) 
## [1] 0.3275134
mae(cases_test_meck,forecast_mod_2_meck$mean) 
## [1] 0.2866063
checkresiduals(wastewater_weather_mod_2_meck)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 11.897, df = 3, p-value = 0.007745
## 
## Model df: 11.   Total lags used: 14
exp(forecast_mod_2_meck$mean[1])
## [1] 3.593441
exp(forecast_mod_2_meck$lower[1,])
##      80%      95% 
## 2.429426 1.974729
exp(forecast_mod_2_meck$upper[1,])
##      80%      95% 
## 5.315172 6.539032
exp(forecast_mod_2_meck$mean[1])-exp(cases_test_meck[1])
## [1] 0.5673137
exp(forecast_mod_2_meck$mean[7])
## [1] 4.669721
exp(forecast_mod_2_meck$lower[7,])
##      80%      95% 
## 2.701036 2.021471
exp(forecast_mod_2_meck$upper[7,])
##       80%       95% 
##  8.073307 10.787340
exp(forecast_mod_2_meck$mean[7])-exp(cases_test_meck[7])
## [1] 0.8152623
exp(forecast_mod_2_meck$mean[14])
## [1] 5.787151
exp(forecast_mod_2_meck$lower[14,])
##      80%      95% 
## 2.516034 1.618894
exp(forecast_mod_2_meck$upper[14,])
##      80%      95% 
## 13.31108 20.68765
exp(forecast_mod_2_meck$mean[14]) -exp(cases_test_meck[14])
## [1] 2.767211
meck_forecast_arimax_weather_plot <- autoplot(forecast_mod_1_meck) + 
  autolayer(forecast_mod_1_meck, series = "Forecasted") +
  autolayer(ts(cases_test_meck,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + 
  xlab("") + ggtitle(NULL) + theme(legend.position = "none") #wake

#new hanover#

precipation_hanover <- xts(full_cases_wastewater_weather_data_hanover$mean_precipation,
                        order.by = full_cases_wastewater_weather_data_hanover$Date)
attr(precipation_hanover,'frequency') <- 7
precipation_hanover <- precipation_hanover[-c(505,506,507)]
precipation_hanover <- as.ts(precipation_hanover)
precipation_hanover
## Time Series:
## Start = c(1, 1) 
## End = c(72, 7) 
## Frequency = 7 
##   [1] 0.0123076923 0.0171428571 0.0840000000 0.0000000000 0.7082352941
##   [6] 0.0533333333 0.0105655160 0.0393333333 0.4750000000 0.0813333333
##  [11] 0.1594444444 0.0328571429 0.1632804587 0.0000000000 0.0000000000
##  [16] 0.0000000000 0.0000000000 0.0023076923 0.0075000000 0.0105655160
##  [21] 0.0000000000 0.1166666667 0.2350000000 0.1732804587 0.5529411765
##  [26] 0.0023076923 0.0000000000 0.2227891611 1.4224405160 0.1640000000
##  [31] 0.0126032171 0.0000000000 0.0193333333 0.0958263680 1.0804762240
##  [36] 0.0186666667 0.0192857143 0.0958823529 0.0729411765 0.1841176471
##  [41] 0.9183333333 0.6694117647 1.1544444444 0.1858823529 0.0000000000
##  [46] 0.0894444444 1.0938888889 0.5237500000 0.0000000000 0.0685714286
##  [51] 0.1637500000 0.0120748754 0.0000000000 0.0078571429 0.0568750000
##  [56] 0.0000000000 0.0132698837 0.0605882353 0.1488888889 0.1325000000
##  [61] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0120748754
##  [66] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0166032171
##  [71] 0.0033333333 0.0633333333 0.7929411765 0.0013333333 0.0056250000
##  [76] 0.0164285714 0.0587500000 0.0119365504 0.0147058824 0.1123529412
##  [81] 0.0000000000 0.0073333333 0.0000000000 0.0332698837 0.1168750000
##  [86] 0.0000000000 0.1184354652 0.7840000000 0.0000000000 0.0000000000
##  [91] 0.0000000000 0.0000000000 0.0000000000 0.0105655160 0.0000000000
##  [96] 0.0000000000 0.0326666667 0.4247058824 0.0012500000 0.0000000000
## [101] 0.0000000000 0.0221428571 0.1594444444 0.0105655160 0.0073333333
## [106] 0.0000000000 0.0012500000 0.0000000000 0.0000000000 0.0000000000
## [111] 0.0000000000 0.0577777778 0.0739365504 0.0000000000 0.0000000000
## [116] 0.0000000000 0.0000000000 0.0000000000 0.0120748754 0.0053333333
## [121] 0.0268750000 0.0130655160 0.0000000000 0.1276470588 0.6204524128
## [126] 0.0000000000 0.0023529412 0.0711111111 0.0755000000 0.2299499082
## [131] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000
## [136] 0.0000000000 0.0005882353 0.0093915698 0.0000000000 0.0084524128
## [141] 0.0047619048 0.1691666667 0.0076840116 0.0000000000 0.0128571429
## [146] 0.0080499170 0.2470833333 0.0005882353 0.0088972766 0.1150000000
## [151] 0.5795619302 3.8220833333 0.9991666667 0.0040909091 0.3095238605
## [156] 0.0947368421 0.0199048256 0.2234782609 0.1947826087 0.3408695652
## [161] 1.1747826087 0.0240476480 0.1403855482 1.2856521739 0.0025000000
## [166] 0.0000000000 0.0010000000 0.6334368807 0.6560455764 0.0447826087
## [171] 0.3436363636 0.0000000000 0.5631619302 0.5033333333 0.0000000000
## [176] 0.0189569768 0.0483333333 0.0086363636 0.0426923077 0.1865018560
## [181] 0.8764285714 0.0000000000 0.0135238605 0.0000000000 0.1409523810
## [186] 0.2792307692 0.5126923077 0.2636000000 0.0104347826 0.0319619302
## [191] 0.1036000000 0.2428770107 0.0213636364 0.1100000000 0.0299567389
## [196] 0.1554166667 0.9557326252 1.5284249329 0.1317391304 0.0314285714
## [201] 0.1141270107 0.0000000000 0.0000000000 0.0371428571 0.2707540213
## [206] 0.3268864714 0.0475000000 0.0000000000 0.0363636364 0.0977272727
## [211] 0.4218864714 0.7177777778 3.3614285714 0.0456802949 0.2051851852
## [216] 0.9453846154 1.8272000000 0.0214286449 0.0079619302 0.0053846154
## [221] 0.0000000000 0.0065018560 0.0000000000 0.0085714286 0.2650000000
## [226] 0.2056000000 1.1503120088 0.0216666667 0.1466137920 1.9556000000
## [231] 0.3349603440 0.2927238605 0.0014814815 0.0635374377 0.0051851852
## [236] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000
## [241] 0.1027258019 0.2346666667 0.0081481481 0.0107326252 0.0062610465
## [246] 0.0020000000 0.1492240095 0.6482758621 0.1356802949 0.3543016085
## [251] 0.0053846154 0.0038461538 0.0053846154 0.0000000000 0.0085185185
## [256] 0.0088000000 0.0316000000 0.0200000000 0.0076840116 0.1560806351
## [261] 1.6235714286 3.6689326985 2.7427891611 0.0157692308 0.0052000000
## [266] 0.0000000000 0.0050000000 0.0065018560 0.0067619302 0.0000000000
## [271] 0.0196153846 0.0137103440 0.0000000000 0.1402918792 0.0162500000
## [276] 0.0958906762 0.0865018560 0.5026923077 0.0763203753 0.1125000000
## [281] 0.0507143653 0.0074603440 0.0071619302 0.0000000000 0.0126557022
## [286] 0.0072727273 0.0040000000 0.0045833333 0.0008000000 0.0000000000
## [291] 0.0000000000 0.0000000000 0.0009523810 0.0005263158 0.0287729428
## [296] 0.2070370370 0.0012000000 0.0231385571 0.2380769231 0.0108695652
## [301] 0.0000000000 0.0000000000 0.0067619302 0.0000000000 0.0004347826
## [306] 0.0000000000 0.0432880122 0.4868000000 0.0133883274 0.0070436773
## [311] 0.0070436773 0.0060000000 0.2466314169 0.0086542720 0.0000000000
## [316] 0.0137729428 0.0000000000 0.0067619302 0.0000000000 0.0041666667
## [321] 0.0000000000 0.0076840116 0.0194331672 0.0583333333 0.0000000000
## [326] 0.0123356312 0.1458333333 0.0404308693 0.0036842105 0.0151346310
## [331] 0.0000000000 0.0076840116 0.0067619302 0.0000000000 0.0000000000
## [336] 0.0000000000 0.0000000000 0.0073499242 0.1236684539 0.8696000000
## [341] 0.0112103440 0.0678260870 0.3156107937 0.0195238372 0.0067619302
## [346] 0.0013043478 0.0004347826 0.0000000000 0.0000000000 0.0665217391
## [351] 0.2788605897 0.0988461538 0.9503703704 0.0125000000 0.0000000000
## [356] 0.0000000000 0.0000000000 0.0000000000 0.0067619302 0.0000000000
## [361] 0.0172000000 0.0164000000 0.0013043478 0.0340909091 0.9176557022
## [366] 0.0379619302 0.0112000000 0.0992307692 0.0012000000 0.0000000000
## [371] 0.0000000000 0.1475000000 0.0000000000 0.0140873547 0.0008333333
## [376] 0.0004347826 0.0000000000 0.3595238095 1.5307619302 0.0076840116
## [381] 0.0070436773 0.0140909091 0.2259090909 0.3220000000 0.0000000000
## [386] 0.0000000000 0.0167316596 0.0158658298 0.0000000000 0.0147165836
## [391] 0.0920000000 0.0546712625 0.0076840116 0.0000000000 0.0004545455
## [396] 0.0421739130 0.0788095483 0.4266666667 0.0080499170 0.0511619302
## [401] 0.2022710868 0.0009523810 0.0000000000 0.0076840116 0.0076840116
## [406] 0.0094736842 0.1019619302 0.0090022979 0.0000000000 0.0043478261
## [411] 0.2084615385 0.0636363636 0.0000000000 0.0065000000 0.0382194894
## [416] 0.0019047619 0.0000000000 0.0000000000 0.0000000000 0.0194444444
## [421] 0.3540909091 0.0000000000 0.0000000000 0.0000000000 0.0000000000
## [426] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.2931818182
## [431] 0.1382194894 0.0142857143 0.2900000000 0.2694444444 0.0088972766
## [436] 0.0000000000 0.0146867503 0.1452380952 0.0088972766 0.0000000000
## [441] 0.0084524128 0.0088972766 0.0084524128 0.0104524128 0.7125000000
## [446] 1.0853770107 0.0005882353 0.0169048256 0.0000000000 0.0000000000
## [451] 0.0179048256 0.0108695652 0.1765384615 0.0000000000 0.0022727273
## [456] 0.0000000000 0.1022294662 0.6380769231 0.4553846154 0.0204545455
## [461] 0.0000000000 0.0076840116 0.0000000000 0.0073499242 0.0000000000
## [466] 0.0000000000 0.0000000000 0.0359524128 0.2143064459 0.3132936773
## [471] 1.0430434783 0.0090022979 0.0000000000 0.0000000000 0.0000000000
## [476] 0.0084524128 0.0000000000 0.0203934024 0.0545833333 0.0076840116
## [481] 0.0000000000 0.0000000000 0.0000000000 0.0080499170 0.0000000000
## [486] 0.0000000000 0.0549603440 0.0013636364 0.0014285714 0.0000000000
## [491] 0.0085261074 0.0000000000 0.0080499170 0.0527272727 0.1811619302
## [496] 0.3216666667 0.2157142857 0.0244524128 0.6560869565 0.0084524128
## [501] 0.0105655160 0.0011111111 0.0000000000 0.0111204857
temp_hanover<- xts(full_cases_wastewater_weather_data_hanover$mean_temp,
                 order.by = full_cases_wastewater_weather_data_hanover$Date)
attr(temp_hanover,'frequency') <- 7
temp_hanover <- temp_hanover[-c(505,506,507)]
temp_hanover <- as.ts(temp_hanover)
temp_hanover
## Time Series:
## Start = c(1, 1) 
## End = c(72, 7) 
## Frequency = 7 
##   [1] 49.33333 42.66667 42.66667 41.00000 44.16667 43.50000 40.66667 41.66667
##   [9] 43.16667 45.83333 47.83333 47.66667 46.00000 40.83333 43.66667 44.00000
##  [17] 46.16667 44.66667 47.66667 44.50000 41.50000 47.16667 58.16667 56.83333
##  [25] 41.16667 37.66667 34.66667 47.83333 45.66667 41.16667 41.00000 39.50000
##  [33] 48.16667 44.16667 42.66667 41.83333 50.16667 54.33333 52.50000 46.50000
##  [41] 44.50000 40.33333 45.16667 56.83333 46.50000 43.33333 38.66667 37.33333
##  [49] 37.83333 47.16667 55.83333 53.66667 59.00000 56.16667 60.50000 68.50000
##  [57] 71.75000 57.50000 49.33333 52.33333 48.50000 44.16667 42.50000 43.33333
##  [65] 50.00000 56.50000 57.16667 61.50000 61.50000 62.00000 57.83333 48.16667
##  [73] 50.16667 61.50000 56.33333 47.50000 52.16667 57.50000 60.66667 65.00000
##  [81] 66.50000 72.00000 70.83333 73.50000 59.50000 57.00000 66.50000 58.50000
##  [89] 43.16667 41.83333 51.83333 61.00000 66.16667 70.33333 72.16667 70.50000
##  [97] 70.33333 71.16667 69.83333 65.33333 62.50000 66.00000 59.50000 59.83333
## [105] 61.66667 62.50000 59.16667 64.83333 52.33333 50.66667 58.33333 62.33333
## [113] 61.00000 63.16667 68.66667 73.66667 71.16667 64.50000 63.50000 71.66667
## [121] 75.83333 77.00000 67.66667 60.33333 60.00000 66.16667 72.16667 66.33333
## [129] 58.33333 57.16667 59.83333 59.83333 62.16667 64.83333 68.33333 68.33333
## [137] 68.33333 66.83333 71.16667 76.83333 79.66667 78.66667 78.16667 84.25000
## [145] 84.00000 79.83333 66.50000 62.16667 66.33333 72.16667 75.16667 77.00000
## [153] 78.50000 78.25000 79.75000 82.25000 82.25000 81.00000 82.50000 81.25000
## [161] 76.25000 78.00000 79.75000 78.75000 77.50000 76.50000 80.00000 81.25000
## [169] 82.25000 80.00000 75.25000 70.50000 73.25000 77.50000 78.75000 79.50000
## [177] 80.25000 81.00000 81.83333 79.83333 76.66667 74.33333 75.50000 76.66667
## [185] 79.50000 80.66667 80.66667 81.33333 82.66667 83.83333 81.16667 81.16667
## [193] 81.66667 83.33333 81.83333 82.00000 78.16667 77.16667 79.83333 82.83333
## [201] 79.33333 75.66667 77.16667 80.00000 83.83333 79.16667 80.00000 83.66667
## [209] 85.83333 84.00000 80.50000 76.50000 73.75000 74.83333 75.00000 77.00000
## [217] 79.50000 78.16667 80.33333 83.16667 84.50000 83.16667 82.66667 82.33333
## [225] 81.50000 80.66667 81.83333 84.50000 81.50000 80.33333 80.25000 79.75000
## [233] 82.50000 82.25000 81.75000 81.00000 81.00000 80.25000 85.00000 83.50000
## [241] 82.00000 76.00000 71.50000 71.50000 72.33333 76.83333 79.16667 79.50000
## [249] 75.83333 71.50000 70.50000 71.50000 74.33333 75.83333 77.16667 78.50000
## [257] 80.66667 78.83333 77.33333 77.50000 76.66667 75.66667 68.16667 68.00000
## [265] 67.66667 68.33333 69.00000 71.66667 77.66667 74.66667 70.83333 69.33333
## [273] 70.50000 73.33333 75.50000 76.16667 74.66667 73.83333 73.50000 72.50000
## [281] 71.33333 69.66667 70.66667 71.33333 72.16667 72.33333 62.66667 58.83333
## [289] 59.83333 63.33333 67.66667 72.16667 67.66667 64.50000 69.50000 66.00000
## [297] 58.33333 59.33333 62.50000 59.66667 61.00000 60.00000 60.33333 56.83333
## [305] 53.16667 48.50000 52.16667 54.33333 54.83333 57.66667 59.33333 60.83333
## [313] 63.83333 58.50000 53.50000 51.50000 51.50000 57.66667 62.33333 55.66667
## [321] 48.50000 54.00000 57.25000 46.50000 40.00000 44.75000 48.75000 45.25000
## [329] 48.00000 49.25000 42.75000 49.83333 56.33333 60.83333 61.16667 59.33333
## [337] 62.66667 59.00000 48.33333 44.83333 52.83333 67.00000 56.00000 49.16667
## [345] 50.66667 52.83333 56.83333 63.83333 67.00000 59.83333 45.16667 43.83333
## [353] 47.33333 44.00000 45.66667 58.50000 64.16667 63.00000 66.66667 70.16667
## [361] 70.83333 69.83333 70.16667 69.33333 58.50000 42.66667 47.00000 51.00000
## [369] 47.50000 38.50000 48.50000 52.83333 37.33333 40.50000 43.50000 46.33333
## [377] 41.33333 45.00000 44.66667 41.16667 43.16667 51.00000 39.83333 30.00000
## [385] 31.50000 38.16667 42.16667 41.33333 36.66667 37.16667 35.50000 30.83333
## [393] 39.00000 42.33333 43.00000 56.83333 66.83333 51.66667 42.00000 44.00000
## [401] 43.66667 42.50000 48.16667 52.50000 55.33333 50.83333 41.50000 40.33333
## [409] 46.50000 57.33333 62.00000 50.83333 44.33333 51.00000 62.16667 69.00000
## [417] 63.00000 65.16667 59.25000 45.50000 47.00000 51.00000 55.50000 64.75000
## [425] 60.00000 57.25000 64.25000 72.00000 68.25000 62.50000 57.50000 51.00000
## [433] 55.75000 43.25000 41.75000 50.75000 59.50000 65.00000 62.50000 67.75000
## [441] 60.25000 53.50000 54.75000 62.50000 68.00000 61.75000 59.00000 54.00000
## [449] 50.00000 47.75000 53.25000 69.25000 65.83333 56.50000 58.66667 56.83333
## [457] 61.00000 70.50000 71.50000 61.00000 56.00000 52.66667 57.66667 69.16667
## [465] 70.16667 72.00000 64.50000 62.00000 65.33333 60.33333 54.00000 52.00000
## [473] 56.00000 63.33333 65.33333 67.50000 69.00000 72.16667 65.16667 59.16667
## [481] 59.00000 66.66667 71.33333 75.00000 78.00000 78.00000 76.00000 76.16667
## [489] 73.16667 60.00000 57.00000 61.00000 65.66667 68.16667 69.16667 69.16667
## [497] 72.50000 77.00000 74.75000 73.00000 79.00000 84.50000 84.25000 80.50000
precipation_hanover_train <- ts(precipation_hanover[-c(491:504)])
precipation_hanover_test <- ts(precipation_hanover[c(491:504)])

temp_hanover_train <- ts(temp_hanover[-c(491:504)])
temp_hanover_test <- ts(temp_hanover[c(491:504)])

vars_hanover <- ts.union(viral_train_hanover,precipation_hanover_train,
                 temp_hanover_train)

vars_test_hanover <- ts.union(viral_test_hanover,precipation_hanover_test,
                      temp_hanover_test)

wastewater_weather_mod_1_hanover <- Arima(cases_train_hanover ,order = c(3,1,1),
                                          xreg = vars_hanover)
coeftest(wastewater_weather_mod_1_hanover) # weakly insignificant
## 
## z test of coefficients:
## 
##                              Estimate  Std. Error z value Pr(>|z|)    
## ar1                       -0.00333870  0.09302964 -0.0359 0.971371    
## ar2                       -0.12057485  0.05821491 -2.0712 0.038340 *  
## ar3                       -0.14248945  0.05319887 -2.6784 0.007397 ** 
## ma1                       -0.50447808  0.08511124 -5.9273 3.08e-09 ***
## viral_train_hanover        0.06713847  0.02623437  2.5592 0.010492 *  
## precipation_hanover_train -0.07448607  0.04217725 -1.7660 0.077392 .  
## temp_hanover_train        -0.00082565  0.00305432 -0.2703 0.786913    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1_hanover <- forecast::forecast(wastewater_weather_mod_1_hanover, h=14,
                                     xreg = vars_test_hanover)
rmse(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.3653023
mae(cases_test_hanover,forecast_mod_1_hanover$mean) 
## [1] 0.3033873
checkresiduals(wastewater_weather_mod_1_hanover)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(3,1,1) errors
## Q* = 32.708, df = 3, p-value = 3.71e-07
## 
## Model df: 7.   Total lags used: 10
wastewater_weather_mod_2_hanover <- Arima(cases_train_hanover ,order = c(4,1,4),
                                       xreg = vars_hanover)
coeftest(wastewater_weather_mod_2_hanover) # weakly insignificant
## 
## z test of coefficients:
## 
##                             Estimate Std. Error  z value  Pr(>|z|)    
## ar1                        0.5978171  0.0618578   9.6644 < 2.2e-16 ***
## ar2                       -0.5058366  0.0346044 -14.6177 < 2.2e-16 ***
## ar3                        1.0158770  0.0356464  28.4987 < 2.2e-16 ***
## ar4                       -0.2217023  0.0588369  -3.7681 0.0001645 ***
## ma1                       -1.1508573  0.0386121 -29.8056 < 2.2e-16 ***
## ma2                        0.6855440  0.0233146  29.4041 < 2.2e-16 ***
## ma3                       -1.2495320  0.0301081 -41.5015 < 2.2e-16 ***
## ma4                        0.8164728  0.0381123  21.4228 < 2.2e-16 ***
## viral_train_hanover        0.0325131  0.0233953   1.3897 0.1646115    
## precipation_hanover_train -0.0586727  0.0413577  -1.4187 0.1559967    
## temp_hanover_train         0.0022913  0.0029263   0.7830 0.4336315    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_2_hanover <- forecast::forecast(wastewater_weather_mod_2_hanover, h=14,
                           xreg = vars_test_hanover)
rmse(cases_test_hanover,forecast_mod_2_hanover$mean) 
## [1] 0.3493521
mae(cases_test_hanover,forecast_mod_2_hanover$mean) 
## [1] 0.250311
checkresiduals(wastewater_weather_mod_2_hanover)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(4,1,4) errors
## Q* = 24.428, df = 3, p-value = 2.033e-05
## 
## Model df: 11.   Total lags used: 14
exp(forecast_mod_2_hanover$mean[1])
## [1] 1.377212
exp(forecast_mod_2_hanover$lower[1,])
##       80%       95% 
## 0.8576265 0.6674302
exp(forecast_mod_2_hanover$upper[1,])
##      80%      95% 
## 2.211583 2.841813
exp(forecast_mod_2_hanover$mean[1])-exp(cases_test_hanover[1])
## [1] 0.269004
exp(forecast_mod_2_hanover$mean[7])
## [1] 2.063698
exp(forecast_mod_2_hanover$lower[7,])
##       80%       95% 
## 1.1079036 0.7970697
exp(forecast_mod_2_hanover$upper[7,])
##      80%      95% 
## 3.844061 5.343133
exp(forecast_mod_2_hanover$mean[7])-exp(cases_test_hanover[7])
## [1] -0.1947093
exp(forecast_mod_2_hanover$mean[14])
## [1] 2.402687
exp(forecast_mod_2_hanover$lower[14,])
##       80%       95% 
## 1.0035497 0.6321565
exp(forecast_mod_2_hanover$upper[14,])
##      80%      95% 
## 5.752484 9.132079
exp(forecast_mod_2_hanover$mean[14]) -exp(cases_test_hanover[14])
## [1] 1.56346
hanover_forecast_arimax_weather_plot <- autoplot(forecast_mod_1_hanover) + 
  autolayer(forecast_mod_1_hanover, series = "Forecasted") +
  autolayer(ts(cases_test_hanover,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + 
  xlab("Time")+ ggtitle(NULL) + theme(legend.position = "bottom") 

SARIMAX- wastewater information only

cases_train_seasonal <- log(cases)[-c(491:504)]
cases_test_seasonal <- log(cases)[c(491:504)]

viral_train_seasonal <- log(viral_gene)[-c(491:504)]
viral_test_seasonal <- log(viral_gene)[c(491:504)]

wastewater_mod_sarimax_wake <- Arima(cases_train_seasonal,
                                    order = c(2,3,2),
                                    seasonal = list(order=c(1,1,2),period=7),
                                    xreg = viral_train_seasonal)
coeftest(wastewater_mod_sarimax_wake)
## 
## z test of coefficients:
## 
##        Estimate Std. Error   z value  Pr(>|z|)    
## ar1  -0.3973303  0.0452715   -8.7766 < 2.2e-16 ***
## ar2  -0.2222987  0.0453107   -4.9061 9.291e-07 ***
## ma1  -1.9646111  0.0107036 -183.5464 < 2.2e-16 ***
## ma2   0.9736454  0.0111200   87.5581 < 2.2e-16 ***
## sar1 -0.9458335  0.0523438  -18.0697 < 2.2e-16 ***
## sma1 -0.0202409  0.0930362   -0.2176    0.8278    
## sma2 -0.9671157  0.0904810  -10.6886 < 2.2e-16 ***
## xreg -0.0027061  0.0231074   -0.1171    0.9068    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_wake <- forecast::forecast(wastewater_mod_sarimax_wake, h=14,
                                 xreg = viral_test_seasonal)
rmse(cases_test_seasonal,forecast_sarimax_wake$mean) 
## [1] 0.1280255
mae(cases_test_seasonal,forecast_sarimax_wake$mean) 
## [1] 0.09187311
checkresiduals(wastewater_mod_sarimax_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 46.219, df = 3, p-value = 5.094e-10
## 
## Model df: 8.   Total lags used: 11
exp(forecast_sarimax_wake$mean[1])
## [1] 3.182921
exp(forecast_sarimax_wake$lower[1,])
##      80%      95% 
## 2.000737 1.564769
exp(forecast_sarimax_wake$upper[1,])
##      80%      95% 
## 5.063627 6.474426
exp(forecast_sarimax_wake$mean[1])-exp(cases_test_seasonal[1])
## [1] -1.232079
exp(forecast_sarimax_wake$mean[7])
## [1] 6.020526
exp(forecast_sarimax_wake$lower[7,])
##      80%      95% 
## 2.299755 1.381754
exp(forecast_sarimax_wake$upper[7,])
##      80%      95% 
## 15.76112 26.23240
exp(forecast_sarimax_wake$mean[7])-exp(cases_test_seasonal[7])
## [1] 0.4755255
exp(forecast_sarimax_wake$mean[14])
## [1] 5.84517
exp(forecast_sarimax_wake$lower[14,])
##       80%       95% 
## 1.0264201 0.4086979
exp(forecast_sarimax_wake$upper[14,])
##      80%      95% 
## 33.28657 83.59722
exp(forecast_sarimax_wake$mean[14]) -exp(cases_test_seasonal[14])
## [1] 1.323503
#mecklenburg

cases_train_meck_seasonal <- log(cases_meck)[-c(491:504)]
cases_test_meck_seasonal <- log(cases_meck)[c(491:504)]

viral_train_meck_seasonal <- log(viral_gene_meck)[-c(491:504)]
viral_test_meck_seasonal <- log(viral_gene_meck)[c(491:504)]

wastewater_mod_sarimax_meck <- Arima(cases_train_meck_seasonal,
                                    order = c(2,3,2),
                                    seasonal = list(order=c(1,1,2),period=7),
                                    xreg = viral_train_meck_seasonal,
                                    method = "CSS")
coeftest(wastewater_mod_sarimax_meck)
## 
## z test of coefficients:
## 
##       Estimate Std. Error   z value  Pr(>|z|)    
## ar1  -0.446571   0.042218  -10.5777 < 2.2e-16 ***
## ar2  -0.285850   0.038450   -7.4344  1.05e-13 ***
## ma1  -1.960809   0.012689 -154.5225 < 2.2e-16 ***
## ma2   0.967986   0.013115   73.8050 < 2.2e-16 ***
## sar1 -0.852261   0.074714  -11.4070 < 2.2e-16 ***
## sma1 -0.078281   0.079329   -0.9868    0.3237    
## sma2 -0.799423   0.075444  -10.5962 < 2.2e-16 ***
## xreg -0.044614   0.030374   -1.4688    0.1419    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_meck <- forecast::forecast(wastewater_mod_sarimax_meck, h=14,
                                 xreg = viral_test_meck_seasonal)
rmse(cases_test_meck_seasonal,forecast_sarimax_meck$mean) 
## [1] 0.1753461
mae(cases_test_meck_seasonal,forecast_sarimax_meck$mean) 
## [1] 0.1423018
checkresiduals(wastewater_mod_sarimax_meck)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 27.449, df = 3, p-value = 4.739e-06
## 
## Model df: 8.   Total lags used: 11
exp(forecast_sarimax_meck$mean[1])
## [1] 2.594116
exp(forecast_sarimax_meck$lower[1,])
##      80%      95% 
## 1.698089 1.356880
exp(forecast_sarimax_meck$upper[1,])
##      80%      95% 
## 3.962949 4.959496
exp(forecast_sarimax_meck$mean[1])-exp(cases_test_meck_seasonal[1])
## [1] 0.4874497
exp(forecast_sarimax_meck$mean[7])
## [1] 2.696436
exp(forecast_sarimax_meck$lower[7,])
##      80%      95% 
## 1.169503 0.751541
exp(forecast_sarimax_meck$upper[7,])
##      80%      95% 
## 6.216972 9.674480
exp(forecast_sarimax_meck$mean[7])-exp(cases_test_meck_seasonal[7])
## [1] -0.09023042
exp(forecast_sarimax_meck$mean[14])
## [1] 2.336227
exp(forecast_sarimax_meck$lower[14,])
##       80%       95% 
## 0.5303645 0.2419317
exp(forecast_sarimax_meck$upper[14,])
##      80%      95% 
## 10.29095 22.55990
exp(forecast_sarimax_meck$mean[14]) -exp(cases_test_meck_seasonal[14])
## [1] 0.1528934
#new hanover

cases_train_hanover_seasonal <- log(cases_hanover)[-c(491:504)]
cases_test_hanover_seasonal <- log(cases_hanover)[c(491:504)]

viral_train_hanover_seasonal <- log(viral_gene_hanover)[-c(491:504)]
viral_test_hanover_seasonal <- log(viral_gene_hanover)[c(491:504)]

wastewater_mod_sarimax_hanover <- Arima(cases_train_hanover_seasonal,
                                    order = c(2,3,2),
                                    seasonal = list(order=c(1,1,2),period=7),
                                    xreg = viral_train_hanover_seasonal,method = "CSS")
coeftest(wastewater_mod_sarimax_hanover)
## 
## z test of coefficients:
## 
##        Estimate Std. Error   z value  Pr(>|z|)    
## ar1  -0.4220079  0.0438132   -9.6320 < 2.2e-16 ***
## ar2  -0.2284375  0.0441468   -5.1745 2.285e-07 ***
## ma1  -1.9683468  0.0093189 -211.2212 < 2.2e-16 ***
## ma2   0.9738603  0.0096201  101.2314 < 2.2e-16 ***
## sar1 -0.0561730  0.0590202   -0.9518   0.34122    
## sma1 -0.7745307  0.0673857  -11.4940 < 2.2e-16 ***
## sma2 -0.1608361  0.0629186   -2.5563   0.01058 *  
## xreg  0.0398994  0.0264399    1.5091   0.13128    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_hanover <- forecast::forecast(wastewater_mod_sarimax_hanover, h=14,
                                 xreg = viral_test_hanover_seasonal)
rmse(cases_test_hanover_seasonal,forecast_sarimax_hanover$mean) 
## [1] 0.3013065
mae(cases_test_hanover_seasonal,forecast_sarimax_hanover$mean) 
## [1] 0.2377347
checkresiduals(wastewater_mod_sarimax_hanover)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 45.095, df = 3, p-value = 8.831e-10
## 
## Model df: 8.   Total lags used: 11
exp(forecast_sarimax_hanover $mean[1])
## [1] 1.125586
exp(forecast_sarimax_hanover $lower[1,])
##       80%       95% 
## 0.6599709 0.4974952
exp(forecast_sarimax_hanover $upper[1,])
##      80%      95% 
## 1.919695 2.546643
exp(forecast_sarimax_hanover $mean[1])-exp(cases_test_hanover_seasonal[1])
## [1] 0.3655855
exp(forecast_sarimax_hanover $mean[7])
## [1] 1.335761
exp(forecast_sarimax_hanover $lower[7,])
##       80%       95% 
## 0.4628380 0.2640956
exp(forecast_sarimax_hanover $upper[7,])
##      80%      95% 
## 3.855035 6.756102
exp(forecast_sarimax_hanover $mean[7])-exp(cases_test_hanover_seasonal[7])
## [1] -0.3192393
exp(forecast_sarimax_hanover $mean[14])
## [1] 1.271932
exp(forecast_sarimax_hanover $lower[14,])
##        80%        95% 
## 0.19129868 0.07017355
exp(forecast_sarimax_hanover $upper[14,])
##       80%       95% 
##  8.456985 23.054412
exp(forecast_sarimax_hanover $mean[14]) -exp(cases_test_hanover_seasonal[14])
## [1] 0.6569316

SARIMAX modelling- Wastewater and weather information

#wake 

vars_weather_wake <- ts.union(viral_train_seasonal,precipation_wake_train,
                 temp_wake_train)

vars_test_weather_wake <- ts.union(viral_test_seasonal,precipation_wake_test,
                      temp_wake_test)

wastewater_weather_mod_sarimax_wake <- Arima(cases_train_seasonal,
                                    order = c(2,3,2),
                                    seasonal = list(order=c(1,1,2),period=7),
                                    xreg = vars_weather_wake)
coeftest(wastewater_weather_mod_sarimax_wake)
## 
## z test of coefficients:
## 
##                          Estimate Std. Error   z value  Pr(>|z|)    
## ar1                    -0.3947980  0.0454890   -8.6790 < 2.2e-16 ***
## ar2                    -0.2171876  0.0455792   -4.7651 1.888e-06 ***
## ma1                    -1.9668423  0.0105821 -185.8652 < 2.2e-16 ***
## ma2                     0.9758908  0.0110089   88.6459 < 2.2e-16 ***
## sar1                   -0.3464399        NaN       NaN       NaN    
## sma1                   -0.5501228        NaN       NaN       NaN    
## sma2                   -0.4497906        NaN       NaN       NaN    
## viral_train_seasonal   -0.0026668  0.0234379   -0.1138    0.9094    
## precipation_wake_train -0.0143747  0.0445356   -0.3228    0.7469    
## temp_wake_train         0.0012620  0.0023898    0.5281    0.5974    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_wake <- forecast::forecast(wastewater_weather_mod_sarimax_wake, h=14,
                                 xreg = vars_test_weather_wake)
rmse(cases_test_seasonal,forecast_sarimax_wake$mean) 
## [1] 0.150083
mae(cases_test_seasonal,forecast_sarimax_wake$mean) 
## [1] 0.1043418
checkresiduals(wastewater_weather_mod_sarimax_wake)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 43.565, df = 3, p-value = 1.867e-09
## 
## Model df: 10.   Total lags used: 13
exp(forecast_sarimax_wake$mean[1])
## [1] 3.006035
exp(forecast_sarimax_wake$lower[1,])
##      80%      95% 
## 1.889125 1.477303
exp(forecast_sarimax_wake$upper[1,])
##      80%      95% 
## 4.783295 6.116716
exp(forecast_sarimax_wake$mean[1])-exp(cases_test_seasonal[1])
## [1] -1.408965
exp(forecast_sarimax_wake$mean[7])
## [1] 5.80686
exp(forecast_sarimax_wake$lower[7,])
##      80%      95% 
## 2.222236 1.336484
exp(forecast_sarimax_wake$upper[7,])
##      80%      95% 
## 15.17374 25.23010
exp(forecast_sarimax_wake$mean[7])-exp(cases_test_seasonal[7])
## [1] 0.26186
exp(forecast_sarimax_wake$mean[14])
## [1] 6.132417
exp(forecast_sarimax_wake$lower[14,])
##       80%       95% 
## 1.0251154 0.3976752
exp(forecast_sarimax_wake$upper[14,])
##      80%      95% 
## 36.68517 94.56596
exp(forecast_sarimax_wake$mean[14]) -exp(cases_test_seasonal[14])
## [1] 1.61075
#mecklenburg

vars_weather_meck <- ts.union(viral_train_meck_seasonal,precipation_meck_train,
                 temp_meck_train)

vars_test_weather_meck <- ts.union(viral_test_meck_seasonal,precipation_meck_test,
                      temp_meck_test)

wastewater_weather_mod_sarimax_meck <- Arima(cases_train_meck_seasonal,
                                    order = c(2,3,2),
                                    seasonal = list(order=c(1,1,2),period=7),
                                    xreg = vars_weather_meck,method = "CSS")
coeftest(wastewater_weather_mod_sarimax_meck)
## 
## z test of coefficients:
## 
##                             Estimate Std. Error   z value  Pr(>|z|)    
## ar1                       -0.4418382  0.0441232  -10.0137 < 2.2e-16 ***
## ar2                       -0.2994553  0.0420548   -7.1206 1.075e-12 ***
## ma1                       -1.9568448  0.0142254 -137.5598 < 2.2e-16 ***
## ma2                        0.9639982  0.0147059   65.5520 < 2.2e-16 ***
## sar1                      -0.7754512  0.1139950   -6.8025 1.028e-11 ***
## sma1                      -0.1508841  0.1176689   -1.2823    0.1997    
## sma2                      -0.7418161  0.1099214   -6.7486 1.493e-11 ***
## viral_train_meck_seasonal -0.0382979  0.0306107   -1.2511    0.2109    
## precipation_meck_train    -0.0460712  0.0446182   -1.0326    0.3018    
## temp_meck_train            0.0032167  0.0024988    1.2873    0.1980    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_meck <- forecast::forecast(wastewater_weather_mod_sarimax_meck, h=14,
                                 xreg = vars_test_weather_meck)
rmse(cases_test_meck_seasonal,forecast_sarimax_meck$mean) 
## [1] 0.2329247
mae(cases_test_meck_seasonal,forecast_sarimax_meck$mean) 
## [1] 0.21248
checkresiduals(wastewater_weather_mod_sarimax_meck)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 30.258, df = 3, p-value = 1.218e-06
## 
## Model df: 10.   Total lags used: 13
exp(forecast_sarimax_meck $mean[1])
## [1] 2.595046
exp(forecast_sarimax_meck $lower[1,])
##      80%      95% 
## 1.699160 1.357931
exp(forecast_sarimax_meck $upper[1,])
##      80%      95% 
## 3.963290 4.959208
exp(forecast_sarimax_meck $mean[1])-exp(cases_test_meck_seasonal[1])
## [1] 0.4883793
exp(forecast_sarimax_meck $mean[7])
## [1] 2.914913
exp(forecast_sarimax_meck $lower[7,])
##       80%       95% 
## 1.2573590 0.8056602
exp(forecast_sarimax_meck $upper[7,])
##       80%       95% 
##  6.757593 10.546283
exp(forecast_sarimax_meck $mean[7])-exp(cases_test_meck_seasonal[7])
## [1] 0.1282467
exp(forecast_sarimax_meck $mean[14])
## [1] 2.751167
exp(forecast_sarimax_meck$lower[14,])
##       80%       95% 
## 0.6101067 0.2748781
exp(forecast_sarimax_meck $upper[14,])
##      80%      95% 
## 12.40590 27.53556
exp(forecast_sarimax_meck $mean[14]) -exp(cases_test_meck_seasonal[14])
## [1] 0.5678341
#new hanover

vars_weather_hanover <- ts.union(viral_train_hanover_seasonal,precipation_hanover_train,
                 temp_hanover_train)

vars_test_weather_hanover <- ts.union(viral_test_hanover_seasonal,precipation_hanover_test,
                      temp_hanover_test)

wastewater_weather_mod_sarimax_hanover <- Arima(cases_train_hanover_seasonal,
                                       order = c(2,3,2),
                                       seasonal = list(order=c(1,1,2),period=7),
                                       xreg = vars_weather_hanover,method = "CSS") #rain is significant
coeftest(wastewater_weather_mod_sarimax_hanover)
## 
## z test of coefficients:
## 
##                                Estimate Std. Error   z value  Pr(>|z|)    
## ar1                          -0.4299720  0.0415283  -10.3537 < 2.2e-16 ***
## ar2                          -0.2308377  0.0444947   -5.1880 2.126e-07 ***
## ma1                          -1.9685358  0.0092815 -212.0919 < 2.2e-16 ***
## ma2                           0.9741181  0.0095571  101.9256 < 2.2e-16 ***
## sar1                          0.0376344  0.1097178    0.3430   0.73159    
## sma1                         -0.8533583  0.1050950   -8.1199 4.667e-16 ***
## sma2                         -0.0872731  0.0982642   -0.8881   0.37446    
## viral_train_hanover_seasonal  0.0403801  0.0262716    1.5370   0.12429    
## precipation_hanover_train    -0.1022082  0.0420563   -2.4303   0.01509 *  
## temp_hanover_train           -0.0010922  0.0026037   -0.4195   0.67487    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_sarimax_hanover <- forecast::forecast(wastewater_weather_mod_sarimax_hanover , h=14,
                                    xreg = vars_test_weather_hanover)
rmse(cases_test_hanover_seasonal,forecast_sarimax_hanover$mean) 
## [1] 0.3003834
mae(cases_test_hanover_seasonal,forecast_sarimax_hanover$mean) 
## [1] 0.2392708
checkresiduals(wastewater_weather_mod_sarimax_hanover)

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(2,3,2)(1,1,2)[7] errors
## Q* = 50.376, df = 3, p-value = 6.645e-11
## 
## Model df: 10.   Total lags used: 13
exp(forecast_sarimax_hanover $mean[1])
## [1] 1.125314
exp(forecast_sarimax_hanover $lower[1,])
##       80%       95% 
## 0.6612874 0.4990775
exp(forecast_sarimax_hanover $upper[1,])
##      80%      95% 
## 1.914948 2.537343
exp(forecast_sarimax_hanover $mean[1])-exp(cases_test_hanover_seasonal[1])
## [1] 0.3653136
exp(forecast_sarimax_hanover $mean[7])
## [1] 1.255077
exp(forecast_sarimax_hanover $lower[7,])
##       80%       95% 
## 0.4391986 0.2519209
exp(forecast_sarimax_hanover $upper[7,])
##      80%      95% 
## 3.586574 6.252829
exp(forecast_sarimax_hanover $mean[7])-exp(cases_test_hanover_seasonal[7])
## [1] -0.3999231
exp(forecast_sarimax_hanover $mean[14])
## [1] 1.187353
exp(forecast_sarimax_hanover $lower[14,])
##        80%        95% 
## 0.17942359 0.06598225
exp(forecast_sarimax_hanover $upper[14,])
##       80%       95% 
##  7.857418 21.366444
exp(forecast_sarimax_hanover $mean[14]) -exp(cases_test_hanover_seasonal[14])
## [1] 0.5723525

Autoregressive Distributed Lag Model

#wake

full_cases_wastewater_weather_data <- full_cases_wastewater_weather_data[-c(505,506,507),]

full_cases_wastewater_weather_data <- full_cases_wastewater_weather_data %>% 
  mutate(log_mean_new_cases = log(mean_new_cases),
         log_viral_gene = log(full_viral_gene_copies_per_person))

full_cases_wastewater_weather_data <- full_cases_wastewater_weather_data %>% 
  mutate(log_mean_new_cases = seasadj(decompose(ts(log_mean_new_cases, frequency=7))),
         log_viral_gene = seasadj(decompose(ts(log_viral_gene, frequency=7))))


full_cases_wastewater_weather_data_train <- 
  full_cases_wastewater_weather_data[-c(491:504),]
full_cases_wastewater_weather_data_test <- 
  full_cases_wastewater_weather_data[c(491:504),]

lowest_rmse <- Inf
best_mod <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                   data = full_cases_wastewater_weather_data_train, p=p,q=q)
    f <- forecast(mod, x= t(full_cases_wastewater_weather_data_test[,7]),h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
                         f$forecasts) #interchanged between RMSE and MAE 
    if (forecast_acc<lowest_rmse){
      lowest_rmse<- forecast_acc
      best_mod <-mod 
    }
  }
}

lowest_rmse #0.209
## [1] 0.2086657
summary(best_mod) #ARDL(1,13)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.57652 -0.16130  0.01015  0.19819  1.58432 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -1.8293899  0.2973165  -6.153 1.65e-09 ***
## log_viral_gene.t      0.0484146  0.0277371   1.745   0.0816 .  
## log_viral_gene.1      0.0085678  0.0351151   0.244   0.8073    
## log_viral_gene.2     -0.0158866  0.0352055  -0.451   0.6520    
## log_viral_gene.3      0.0315251  0.0352034   0.896   0.3710    
## log_viral_gene.4     -0.0005125  0.0353556  -0.014   0.9884    
## log_viral_gene.5      0.0488269  0.0354301   1.378   0.1688    
## log_viral_gene.6      0.0044543  0.0354949   0.125   0.9002    
## log_viral_gene.7     -0.0420197  0.0354797  -1.184   0.2369    
## log_viral_gene.8      0.0078008  0.0354309   0.220   0.8258    
## log_viral_gene.9      0.0243878  0.0353034   0.691   0.4900    
## log_viral_gene.10     0.0041653  0.0351960   0.118   0.9058    
## log_viral_gene.11     0.0531482  0.0352077   1.510   0.1318    
## log_viral_gene.12    -0.0761592  0.0351809  -2.165   0.0309 *  
## log_viral_gene.13     0.0366936  0.0273371   1.342   0.1802    
## log_mean_new_cases.1  0.7723686  0.0305033  25.321  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3619 on 461 degrees of freedom
## Multiple R-squared:  0.8974, Adjusted R-squared:  0.8941 
## F-statistic: 268.9 on 15 and 461 DF,  p-value: < 2.2e-16
checkresiduals(best_mod)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
## -0.0417420903 -0.1369120561 -0.0811557848  0.0435192023 -0.0016408437 
##            19            20            21            22            23 
## -0.0466036986  0.1573381802 -0.0806377566 -0.1933854745  0.3101505602 
##            24            25            26            27            28 
## -0.1080494203 -0.0028412471 -0.2665604050  0.4021827208 -0.2839760165 
##            29            30            31            32            33 
## -0.2712864079  0.3290206437 -0.1017006024  0.1305949022  0.1668407987 
##            34            35            36            37            38 
##  0.0448190875 -0.2163680309 -0.2750283404  0.2805826917 -0.1613045822 
##            39            40            41            42            43 
##  0.0237548319  0.0480230126 -0.0337460028 -0.3416568330 -0.0463296676 
##            44            45            46            47            48 
## -0.0866027853  0.1089414939 -0.0324475677 -0.1849449296  0.0716646630 
##            49            50            51            52            53 
## -0.3009231769  0.1761465237 -0.1845748377 -0.0376955632 -0.3678322292 
##            54            55            56            57            58 
##  0.0501868298  0.0747823443  0.0241326923  0.1356912746 -0.1475208265 
##            59            60            61            62            63 
## -0.2167145002  0.2368083017  0.1929601982 -0.0615718762  0.1445494422 
##            64            65            66            67            68 
##  0.0673490526  0.2318447961 -0.0391711425  0.0989738831  0.2611055830 
##            69            70            71            72            73 
##  0.0607598618  0.2906521259  0.3758893647 -0.0947328765 -0.0233176906 
##            74            75            76            77            78 
##  0.2258300663 -0.2052906808  0.3033834899 -0.0742531265  0.2128799418 
##            79            80            81            82            83 
## -0.0012326035 -0.0569668744 -0.3010897423  0.0983904446 -0.1307668221 
##            84            85            86            87            88 
##  0.1301565258  0.2880814023 -0.4506209383 -0.0339348136  0.2310911786 
##            89            90            91            92            93 
## -0.2921921745 -0.3837856584  0.0510435751  0.5398327032  0.3230214154 
##            94            95            96            97            98 
## -0.0111990534 -0.0154002712  0.3140009465  0.1365612275  0.2043764052 
##            99           100           101           102           103 
##  0.3393487500 -0.4196847501  0.2239302083 -0.1418005133  0.0734271888 
##           104           105           106           107           108 
## -0.0031750457  0.0347172298  0.0626419069  0.0342815316 -0.1211320123 
##           109           110           111           112           113 
##  0.0541302968  0.1666334451 -0.0399103753  0.2543234763  0.2456179424 
##           114           115           116           117           118 
## -0.0231691384  0.0548536211  0.0897910387 -0.0187026153  0.2036108366 
##           119           120           121           122           123 
## -0.1938760213  0.4699144839 -0.3238426089  0.3118557660 -0.0514199559 
##           124           125           126           127           128 
##  0.1991516566 -0.0507167072  0.1376719697  0.3230301430  0.0255712962 
##           129           130           131           132           133 
##  0.2578087262  0.3273859857  0.0177462481  0.3556702852  0.0545730849 
##           134           135           136           137           138 
##  0.2748981049  0.1191885822  0.5697755582 -0.1110192663  0.3421205998 
##           139           140           141           142           143 
## -0.1226349489  0.4226971580  0.4120763796 -0.1490102836  0.2916516294 
##           144           145           146           147           148 
## -0.2427905719  0.0074471904 -0.0321244679  0.1161896650  0.3543573614 
##           149           150           151           152           153 
## -1.4051591731  0.4609759708 -1.2150272598  0.4569726822  0.3660877872 
##           154           155           156           157           158 
##  0.0736287475  0.2354941145 -0.7803499279 -0.1869356537 -0.1567051848 
##           159           160           161           162           163 
## -0.0228174161 -0.4698843322  0.1772091018 -0.2565636090  0.3835435963 
##           164           165           166           167           168 
## -0.0875063418  0.0141112213 -0.0932070545  0.2068339998 -0.1245688146 
##           169           170           171           172           173 
## -0.2461291335 -0.2106265673  0.1337901606 -0.7539476665  0.2267374761 
##           174           175           176           177           178 
## -0.3671512347  0.3398769865 -0.2095862734 -0.2500480979 -0.3606676615 
##           179           180           181           182           183 
## -0.1927440440  0.0715092358 -0.0129852443 -0.3684919557 -1.8022742628 
##           184           185           186           187           188 
##  0.9374304282 -0.1042708185 -0.1418902838 -0.1578875076  0.2557876255 
##           189           190           191           192           193 
## -0.1564654522  0.0502398243 -0.0111697088  0.0995915432 -0.3831805874 
##           194           195           196           197           198 
##  0.3732083778 -0.2070252711  0.0054438357  0.2273607948 -0.1851797459 
##           199           200           201           202           203 
##  0.4965422249 -0.1820921641  0.3835561902 -0.0174186630  0.1186078126 
##           204           205           206           207           208 
##  0.4620427073 -0.0538773460 -0.0073503477 -0.0377942311  0.4212510944 
##           209           210           211           212           213 
##  0.0351436843  0.0143221089  0.0133230216  0.1506924911 -0.0732146581 
##           214           215           216           217           218 
##  0.0565770077  0.2923077294 -0.1968382447  0.0720834037  0.1827291257 
##           219           220           221           222           223 
##  0.0163983116  0.0065368839 -0.0161041785  0.2298531512 -0.0141333796 
##           224           225           226           227           228 
## -0.0356840784  0.0620217030  0.1725118143 -0.0290619715  0.1738873934 
##           229           230           231           232           233 
##  0.1744328708  0.0588796280  0.1473922573 -0.0506241753  0.3650467301 
##           234           235           236           237           238 
##  0.0105123048  0.2391564876  0.2025762708  0.1782099655  0.6066925814 
##           239           240           241           242           243 
## -0.4699064989  0.1982762473 -0.0637070384  0.3247311781  0.3362062436 
##           244           245           246           247           248 
## -0.0653722090 -0.0124316783 -0.6368354247 -1.4071837325  1.3918157282 
##           249           250           251           252           253 
## -0.3688846272 -0.2060525205 -0.0441966021 -0.0088615750 -0.3701163671 
##           254           255           256           257           258 
##  0.0099508823 -0.3011100906  0.0540678414 -0.0593648808 -0.1376819899 
##           259           260           261           262           263 
##  0.0174208465 -0.3841236158  0.0165630297 -0.1778271842 -0.1683936105 
##           264           265           266           267           268 
## -0.2007279909  0.0365810160 -0.3377930814 -0.4799853623  0.4156396119 
##           269           270           271           272           273 
## -0.3238152079 -0.1231902060 -0.0585318447 -0.1125978653  0.1281962132 
##           274           275           276           277           278 
## -0.5899374081 -0.0205667960 -0.2712944676  0.0248145355 -0.1296260118 
##           279           280           281           282           283 
##  0.6444936511 -0.8025320281  0.4452901692 -0.3820986390  0.0747517168 
##           284           285           286           287           288 
##  0.4926687499 -0.1494903907 -0.4257235357 -0.2404533865 -0.6329416583 
##           289           290           291           292           293 
##  0.5813457079 -0.1929994430  0.0922662955 -2.5765239843  0.1028463117 
##           294           295           296           297           298 
##  1.5843169977 -0.7332322413  0.1456505085  0.0880149229  0.0350335481 
##           299           300           301           302           303 
## -0.1417491502  0.1228814549 -0.0076348383 -0.5264663534 -0.0489343167 
##           304           305           306           307           308 
## -0.1303017818  0.0003090705 -0.3466866660  0.1320941090 -0.1133320864 
##           309           310           311           312           313 
##  0.2663365374 -0.4535447855 -0.9682153034  0.7374317924 -0.4911336400 
##           314           315           316           317           318 
##  0.4894482864 -0.4021660159  0.1336820654  0.0158665000 -0.2699853463 
##           319           320           321           322           323 
##  0.2513521206 -0.0157973882  0.1833177813  0.4975719801 -0.0086839980 
##           324           325           326           327           328 
##  0.1040976013 -0.0183429175  0.0414719477 -0.5951589103  0.4807385122 
##           329           330           331           332           333 
##  0.3470406853 -0.1866525809  0.4680886781  0.0319947060  0.5325468267 
##           334           335           336           337           338 
## -0.0905505713  0.1891315399  0.1557188900  0.2388000069 -0.1002797472 
##           339           340           341           342           343 
##  0.1557599655 -0.0921263945 -0.2667844223  0.3283687195 -0.0496379808 
##           344           345           346           347           348 
##  0.2058769375 -0.1141916360 -0.2235378751 -0.0241232474  0.0680237209 
##           349           350           351           352           353 
##  0.1376601310  0.3572567524  0.0089015033 -0.0613286925  0.2305173265 
##           354           355           356           357           358 
##  0.2813167444  0.5406544581 -0.3814566932 -0.3608459192  1.1521768553 
##           359           360           361           362           363 
##  0.7330512757  0.5187172790  0.2947534836  0.4130223035 -0.1119429921 
##           364           365           366           367           368 
##  0.3311350218  0.3317737711  0.5040590744  0.6056026593  0.2473646271 
##           369           370           371           372           373 
##  0.2938392620  0.1981854549  0.2290407568 -0.3467675258  0.7320017788 
##           374           375           376           377           378 
##  0.0821233353  0.2723873311  0.1887300647  0.1205609116 -0.1459035478 
##           379           380           381           382           383 
## -1.2583151036  0.6377162315  0.6372681151  0.0013106804  0.0064903769 
##           384           385           386           387           388 
## -1.0329973511 -0.7159866741  0.8742278542  0.5515743621 -0.2028719947 
##           389           390           391           392           393 
## -0.2218632780 -0.0650463132  0.0101539147 -0.5810336463  0.1329577554 
##           394           395           396           397           398 
##  0.3760437809 -0.2660272083 -0.1108463155 -0.0367582966  0.1458798996 
##           399           400           401           402           403 
## -0.0052593892  0.0578327783 -0.1981317786 -0.1496768479  0.1474334973 
##           404           405           406           407           408 
## -0.2485770733 -0.1061939044  0.0232850213 -0.0074653224  0.0189484295 
##           409           410           411           412           413 
## -0.2056837937 -0.1757078516  0.1379894835 -0.2462304676 -0.1889486846 
##           414           415           416           417           418 
##  0.2558817230 -0.3026764512 -0.2437830183 -0.2815253453 -0.0914274160 
##           419           420           421           422           423 
##  0.0105115325 -0.6405113615  0.2059791118 -0.4056586806 -0.2603566237 
##           424           425           426           427           428 
##  0.0606420781  0.0429674664 -0.7664714500  0.0744346446  0.1809520782 
##           429           430           431           432           433 
##  0.0273559425  0.0502766821 -0.5315485962  0.4725405057 -0.3414037353 
##           434           435           436           437           438 
##  0.0216919893 -0.2517964430  0.2858718821 -0.5545715262  0.2511257348 
##           439           440           441           442           443 
## -0.1602194917 -0.2850642895 -0.0718603947  0.2730619769  0.0090388626 
##           444           445           446           447           448 
## -0.5912043247  0.1507535115  0.2244142512 -0.2906564518 -0.4290569024 
##           449           450           451           452           453 
## -0.0286620474 -0.3903306490  0.2573945926 -0.4391753938 -0.0449418303 
##           454           455           456           457           458 
##  0.3557557140 -0.3917795385  0.1484321931 -0.1296784418 -0.3206768038 
##           459           460           461           462           463 
##  0.3131271331 -0.2010198021  0.0828836570  0.1902485802 -0.2157183408 
##           464           465           466           467           468 
## -0.2235392099  0.2295261591  0.2135475376 -0.0092074661 -0.1650120848 
##           469           470           471           472           473 
##  0.3170934123 -0.1158619491 -0.0368124692  0.1904676658 -0.1610594475 
##           474           475           476           477           478 
##  0.0178717919  0.0976490303  0.0330803268  0.2252708562 -0.3824507278 
##           479           480           481           482           483 
##  0.0827837494  0.0415253667 -0.1986518520  0.1385059049 -0.0228448686 
##           484           485           486           487           488 
## -0.2786497766  0.1569772924 -0.0941576332  0.0169421430  0.1704528206 
##           489           490 
## -0.1485905883  0.2629694349

mod_ardl113 <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
               data = full_cases_wastewater_weather_data_train, p=13,q=1)
summary(mod_ardl113) 
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.57652 -0.16130  0.01015  0.19819  1.58432 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -1.8293899  0.2973165  -6.153 1.65e-09 ***
## log_viral_gene.t      0.0484146  0.0277371   1.745   0.0816 .  
## log_viral_gene.1      0.0085678  0.0351151   0.244   0.8073    
## log_viral_gene.2     -0.0158866  0.0352055  -0.451   0.6520    
## log_viral_gene.3      0.0315251  0.0352034   0.896   0.3710    
## log_viral_gene.4     -0.0005125  0.0353556  -0.014   0.9884    
## log_viral_gene.5      0.0488269  0.0354301   1.378   0.1688    
## log_viral_gene.6      0.0044543  0.0354949   0.125   0.9002    
## log_viral_gene.7     -0.0420197  0.0354797  -1.184   0.2369    
## log_viral_gene.8      0.0078008  0.0354309   0.220   0.8258    
## log_viral_gene.9      0.0243878  0.0353034   0.691   0.4900    
## log_viral_gene.10     0.0041653  0.0351960   0.118   0.9058    
## log_viral_gene.11     0.0531482  0.0352077   1.510   0.1318    
## log_viral_gene.12    -0.0761592  0.0351809  -2.165   0.0309 *  
## log_viral_gene.13     0.0366936  0.0273371   1.342   0.1802    
## log_mean_new_cases.1  0.7723686  0.0305033  25.321  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3619 on 461 degrees of freedom
## Multiple R-squared:  0.8974, Adjusted R-squared:  0.8941 
## F-statistic: 268.9 on 15 and 461 DF,  p-value: < 2.2e-16
f_ardl113  <- forecast(mod_ardl113, 
                       x= t(full_cases_wastewater_weather_data_test[,7]),
                       h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl113$forecasts)
## [1] 0.2086657
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl113$forecasts)
## [1] 0.1920911
checkresiduals(mod_ardl113)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
## -0.0417420903 -0.1369120561 -0.0811557848  0.0435192023 -0.0016408437 
##            19            20            21            22            23 
## -0.0466036986  0.1573381802 -0.0806377566 -0.1933854745  0.3101505602 
##            24            25            26            27            28 
## -0.1080494203 -0.0028412471 -0.2665604050  0.4021827208 -0.2839760165 
##            29            30            31            32            33 
## -0.2712864079  0.3290206437 -0.1017006024  0.1305949022  0.1668407987 
##            34            35            36            37            38 
##  0.0448190875 -0.2163680309 -0.2750283404  0.2805826917 -0.1613045822 
##            39            40            41            42            43 
##  0.0237548319  0.0480230126 -0.0337460028 -0.3416568330 -0.0463296676 
##            44            45            46            47            48 
## -0.0866027853  0.1089414939 -0.0324475677 -0.1849449296  0.0716646630 
##            49            50            51            52            53 
## -0.3009231769  0.1761465237 -0.1845748377 -0.0376955632 -0.3678322292 
##            54            55            56            57            58 
##  0.0501868298  0.0747823443  0.0241326923  0.1356912746 -0.1475208265 
##            59            60            61            62            63 
## -0.2167145002  0.2368083017  0.1929601982 -0.0615718762  0.1445494422 
##            64            65            66            67            68 
##  0.0673490526  0.2318447961 -0.0391711425  0.0989738831  0.2611055830 
##            69            70            71            72            73 
##  0.0607598618  0.2906521259  0.3758893647 -0.0947328765 -0.0233176906 
##            74            75            76            77            78 
##  0.2258300663 -0.2052906808  0.3033834899 -0.0742531265  0.2128799418 
##            79            80            81            82            83 
## -0.0012326035 -0.0569668744 -0.3010897423  0.0983904446 -0.1307668221 
##            84            85            86            87            88 
##  0.1301565258  0.2880814023 -0.4506209383 -0.0339348136  0.2310911786 
##            89            90            91            92            93 
## -0.2921921745 -0.3837856584  0.0510435751  0.5398327032  0.3230214154 
##            94            95            96            97            98 
## -0.0111990534 -0.0154002712  0.3140009465  0.1365612275  0.2043764052 
##            99           100           101           102           103 
##  0.3393487500 -0.4196847501  0.2239302083 -0.1418005133  0.0734271888 
##           104           105           106           107           108 
## -0.0031750457  0.0347172298  0.0626419069  0.0342815316 -0.1211320123 
##           109           110           111           112           113 
##  0.0541302968  0.1666334451 -0.0399103753  0.2543234763  0.2456179424 
##           114           115           116           117           118 
## -0.0231691384  0.0548536211  0.0897910387 -0.0187026153  0.2036108366 
##           119           120           121           122           123 
## -0.1938760213  0.4699144839 -0.3238426089  0.3118557660 -0.0514199559 
##           124           125           126           127           128 
##  0.1991516566 -0.0507167072  0.1376719697  0.3230301430  0.0255712962 
##           129           130           131           132           133 
##  0.2578087262  0.3273859857  0.0177462481  0.3556702852  0.0545730849 
##           134           135           136           137           138 
##  0.2748981049  0.1191885822  0.5697755582 -0.1110192663  0.3421205998 
##           139           140           141           142           143 
## -0.1226349489  0.4226971580  0.4120763796 -0.1490102836  0.2916516294 
##           144           145           146           147           148 
## -0.2427905719  0.0074471904 -0.0321244679  0.1161896650  0.3543573614 
##           149           150           151           152           153 
## -1.4051591731  0.4609759708 -1.2150272598  0.4569726822  0.3660877872 
##           154           155           156           157           158 
##  0.0736287475  0.2354941145 -0.7803499279 -0.1869356537 -0.1567051848 
##           159           160           161           162           163 
## -0.0228174161 -0.4698843322  0.1772091018 -0.2565636090  0.3835435963 
##           164           165           166           167           168 
## -0.0875063418  0.0141112213 -0.0932070545  0.2068339998 -0.1245688146 
##           169           170           171           172           173 
## -0.2461291335 -0.2106265673  0.1337901606 -0.7539476665  0.2267374761 
##           174           175           176           177           178 
## -0.3671512347  0.3398769865 -0.2095862734 -0.2500480979 -0.3606676615 
##           179           180           181           182           183 
## -0.1927440440  0.0715092358 -0.0129852443 -0.3684919557 -1.8022742628 
##           184           185           186           187           188 
##  0.9374304282 -0.1042708185 -0.1418902838 -0.1578875076  0.2557876255 
##           189           190           191           192           193 
## -0.1564654522  0.0502398243 -0.0111697088  0.0995915432 -0.3831805874 
##           194           195           196           197           198 
##  0.3732083778 -0.2070252711  0.0054438357  0.2273607948 -0.1851797459 
##           199           200           201           202           203 
##  0.4965422249 -0.1820921641  0.3835561902 -0.0174186630  0.1186078126 
##           204           205           206           207           208 
##  0.4620427073 -0.0538773460 -0.0073503477 -0.0377942311  0.4212510944 
##           209           210           211           212           213 
##  0.0351436843  0.0143221089  0.0133230216  0.1506924911 -0.0732146581 
##           214           215           216           217           218 
##  0.0565770077  0.2923077294 -0.1968382447  0.0720834037  0.1827291257 
##           219           220           221           222           223 
##  0.0163983116  0.0065368839 -0.0161041785  0.2298531512 -0.0141333796 
##           224           225           226           227           228 
## -0.0356840784  0.0620217030  0.1725118143 -0.0290619715  0.1738873934 
##           229           230           231           232           233 
##  0.1744328708  0.0588796280  0.1473922573 -0.0506241753  0.3650467301 
##           234           235           236           237           238 
##  0.0105123048  0.2391564876  0.2025762708  0.1782099655  0.6066925814 
##           239           240           241           242           243 
## -0.4699064989  0.1982762473 -0.0637070384  0.3247311781  0.3362062436 
##           244           245           246           247           248 
## -0.0653722090 -0.0124316783 -0.6368354247 -1.4071837325  1.3918157282 
##           249           250           251           252           253 
## -0.3688846272 -0.2060525205 -0.0441966021 -0.0088615750 -0.3701163671 
##           254           255           256           257           258 
##  0.0099508823 -0.3011100906  0.0540678414 -0.0593648808 -0.1376819899 
##           259           260           261           262           263 
##  0.0174208465 -0.3841236158  0.0165630297 -0.1778271842 -0.1683936105 
##           264           265           266           267           268 
## -0.2007279909  0.0365810160 -0.3377930814 -0.4799853623  0.4156396119 
##           269           270           271           272           273 
## -0.3238152079 -0.1231902060 -0.0585318447 -0.1125978653  0.1281962132 
##           274           275           276           277           278 
## -0.5899374081 -0.0205667960 -0.2712944676  0.0248145355 -0.1296260118 
##           279           280           281           282           283 
##  0.6444936511 -0.8025320281  0.4452901692 -0.3820986390  0.0747517168 
##           284           285           286           287           288 
##  0.4926687499 -0.1494903907 -0.4257235357 -0.2404533865 -0.6329416583 
##           289           290           291           292           293 
##  0.5813457079 -0.1929994430  0.0922662955 -2.5765239843  0.1028463117 
##           294           295           296           297           298 
##  1.5843169977 -0.7332322413  0.1456505085  0.0880149229  0.0350335481 
##           299           300           301           302           303 
## -0.1417491502  0.1228814549 -0.0076348383 -0.5264663534 -0.0489343167 
##           304           305           306           307           308 
## -0.1303017818  0.0003090705 -0.3466866660  0.1320941090 -0.1133320864 
##           309           310           311           312           313 
##  0.2663365374 -0.4535447855 -0.9682153034  0.7374317924 -0.4911336400 
##           314           315           316           317           318 
##  0.4894482864 -0.4021660159  0.1336820654  0.0158665000 -0.2699853463 
##           319           320           321           322           323 
##  0.2513521206 -0.0157973882  0.1833177813  0.4975719801 -0.0086839980 
##           324           325           326           327           328 
##  0.1040976013 -0.0183429175  0.0414719477 -0.5951589103  0.4807385122 
##           329           330           331           332           333 
##  0.3470406853 -0.1866525809  0.4680886781  0.0319947060  0.5325468267 
##           334           335           336           337           338 
## -0.0905505713  0.1891315399  0.1557188900  0.2388000069 -0.1002797472 
##           339           340           341           342           343 
##  0.1557599655 -0.0921263945 -0.2667844223  0.3283687195 -0.0496379808 
##           344           345           346           347           348 
##  0.2058769375 -0.1141916360 -0.2235378751 -0.0241232474  0.0680237209 
##           349           350           351           352           353 
##  0.1376601310  0.3572567524  0.0089015033 -0.0613286925  0.2305173265 
##           354           355           356           357           358 
##  0.2813167444  0.5406544581 -0.3814566932 -0.3608459192  1.1521768553 
##           359           360           361           362           363 
##  0.7330512757  0.5187172790  0.2947534836  0.4130223035 -0.1119429921 
##           364           365           366           367           368 
##  0.3311350218  0.3317737711  0.5040590744  0.6056026593  0.2473646271 
##           369           370           371           372           373 
##  0.2938392620  0.1981854549  0.2290407568 -0.3467675258  0.7320017788 
##           374           375           376           377           378 
##  0.0821233353  0.2723873311  0.1887300647  0.1205609116 -0.1459035478 
##           379           380           381           382           383 
## -1.2583151036  0.6377162315  0.6372681151  0.0013106804  0.0064903769 
##           384           385           386           387           388 
## -1.0329973511 -0.7159866741  0.8742278542  0.5515743621 -0.2028719947 
##           389           390           391           392           393 
## -0.2218632780 -0.0650463132  0.0101539147 -0.5810336463  0.1329577554 
##           394           395           396           397           398 
##  0.3760437809 -0.2660272083 -0.1108463155 -0.0367582966  0.1458798996 
##           399           400           401           402           403 
## -0.0052593892  0.0578327783 -0.1981317786 -0.1496768479  0.1474334973 
##           404           405           406           407           408 
## -0.2485770733 -0.1061939044  0.0232850213 -0.0074653224  0.0189484295 
##           409           410           411           412           413 
## -0.2056837937 -0.1757078516  0.1379894835 -0.2462304676 -0.1889486846 
##           414           415           416           417           418 
##  0.2558817230 -0.3026764512 -0.2437830183 -0.2815253453 -0.0914274160 
##           419           420           421           422           423 
##  0.0105115325 -0.6405113615  0.2059791118 -0.4056586806 -0.2603566237 
##           424           425           426           427           428 
##  0.0606420781  0.0429674664 -0.7664714500  0.0744346446  0.1809520782 
##           429           430           431           432           433 
##  0.0273559425  0.0502766821 -0.5315485962  0.4725405057 -0.3414037353 
##           434           435           436           437           438 
##  0.0216919893 -0.2517964430  0.2858718821 -0.5545715262  0.2511257348 
##           439           440           441           442           443 
## -0.1602194917 -0.2850642895 -0.0718603947  0.2730619769  0.0090388626 
##           444           445           446           447           448 
## -0.5912043247  0.1507535115  0.2244142512 -0.2906564518 -0.4290569024 
##           449           450           451           452           453 
## -0.0286620474 -0.3903306490  0.2573945926 -0.4391753938 -0.0449418303 
##           454           455           456           457           458 
##  0.3557557140 -0.3917795385  0.1484321931 -0.1296784418 -0.3206768038 
##           459           460           461           462           463 
##  0.3131271331 -0.2010198021  0.0828836570  0.1902485802 -0.2157183408 
##           464           465           466           467           468 
## -0.2235392099  0.2295261591  0.2135475376 -0.0092074661 -0.1650120848 
##           469           470           471           472           473 
##  0.3170934123 -0.1158619491 -0.0368124692  0.1904676658 -0.1610594475 
##           474           475           476           477           478 
##  0.0178717919  0.0976490303  0.0330803268  0.2252708562 -0.3824507278 
##           479           480           481           482           483 
##  0.0827837494  0.0415253667 -0.1986518520  0.1385059049 -0.0228448686 
##           484           485           486           487           488 
## -0.2786497766  0.1569772924 -0.0941576332  0.0169421430  0.1704528206 
##           489           490 
## -0.1485905883  0.2629694349

mod_ardl1411 <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                       data = full_cases_wastewater_weather_data_train, 
                       p=11,q=14)
summary(mod_ardl1411) 
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4613 -0.1434  0.0140  0.1672  1.3165 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -0.935982   0.311323  -3.006 0.002791 ** 
## log_viral_gene.t       0.020582   0.026380   0.780 0.435674    
## log_viral_gene.1       0.014493   0.033027   0.439 0.661012    
## log_viral_gene.2      -0.019663   0.033113  -0.594 0.552942    
## log_viral_gene.3       0.023124   0.033200   0.696 0.486477    
## log_viral_gene.4       0.003947   0.033226   0.119 0.905501    
## log_viral_gene.5       0.027756   0.033143   0.837 0.402769    
## log_viral_gene.6       0.011458   0.033188   0.345 0.730062    
## log_viral_gene.7      -0.029281   0.033204  -0.882 0.378326    
## log_viral_gene.8      -0.008775   0.033144  -0.265 0.791331    
## log_viral_gene.9       0.017917   0.033017   0.543 0.587637    
## log_viral_gene.10     -0.005994   0.032735  -0.183 0.854791    
## log_viral_gene.11      0.013394   0.026259   0.510 0.610240    
## log_mean_new_cases.1   0.468930   0.047161   9.943  < 2e-16 ***
## log_mean_new_cases.2   0.109227   0.052173   2.094 0.036860 *  
## log_mean_new_cases.3   0.080004   0.052447   1.525 0.127855    
## log_mean_new_cases.4   0.110467   0.052580   2.101 0.036205 *  
## log_mean_new_cases.5   0.177483   0.053019   3.348 0.000884 ***
## log_mean_new_cases.6   0.036936   0.053668   0.688 0.491663    
## log_mean_new_cases.7   0.119539   0.053589   2.231 0.026196 *  
## log_mean_new_cases.8  -0.083050   0.053541  -1.551 0.121570    
## log_mean_new_cases.9  -0.039005   0.053626  -0.727 0.467391    
## log_mean_new_cases.10  0.031823   0.053127   0.599 0.549476    
## log_mean_new_cases.11 -0.017356   0.052915  -0.328 0.743074    
## log_mean_new_cases.12 -0.084198   0.052775  -1.595 0.111326    
## log_mean_new_cases.13  0.033950   0.052741   0.644 0.520085    
## log_mean_new_cases.14 -0.073242   0.047306  -1.548 0.122268    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3357 on 449 degrees of freedom
## Multiple R-squared:  0.9139, Adjusted R-squared:  0.9089 
## F-statistic: 183.4 on 26 and 449 DF,  p-value: < 2.2e-16
f_ardl1411  <- forecast(mod_ardl1411, 
                        x= t(full_cases_wastewater_weather_data_test[,7]),
                        h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl1411$forecasts)
## [1] 0.2168104
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
    f_ardl1411$forecasts)
## [1] 0.1836546
checkresiduals(mod_ardl1411)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
## -0.1842245166 -0.0861296050 -0.0262578456 -0.0064723464 -0.0426762850 
##            20            21            22            23            24 
##  0.1811373604  0.0175349617 -0.1572505301  0.2730325964 -0.0232214417 
##            25            26            27            28            29 
##  0.0115445452 -0.2293143393  0.2873347852 -0.2672939781 -0.3355455099 
##            30            31            32            33            34 
##  0.1390014025 -0.0564782417  0.0875242762  0.2352770838  0.1152006941 
##            35            36            37            38            39 
## -0.1010711348 -0.2766170270  0.1352443894 -0.2188829882  0.0464003751 
##            40            41            42            43            44 
##  0.0316241672  0.0445627290 -0.2286396830 -0.0865399301 -0.1053300342 
##            45            46            47            48            49 
##  0.1426812352  0.0857694957 -0.0791202898  0.0996618059 -0.1839208459 
##            50            51            52            53            54 
##  0.0674317683 -0.1513743867 -0.0636848500 -0.3993395225 -0.0339924676 
##            55            56            57            58            59 
##  0.0336237046  0.0765350679  0.1969309729  0.0005172382 -0.2582453244 
##            60            61            62            63            64 
##  0.1758497019  0.1145688825 -0.1028181764  0.0807199478  0.0316316219 
##            65            66            67            68            69 
##  0.1327846706 -0.0183215964  0.0226460974  0.2157107123  0.1416170896 
##            70            71            72            73            74 
##  0.2296614908  0.5055091488  0.0521561970 -0.0974273217  0.0682491883 
##            75            76            77            78            79 
## -0.3464081267  0.0886740656 -0.0443016793  0.0716764279  0.0534669820 
##            80            81            82            83            84 
##  0.0212748623 -0.3266746536  0.0637852691 -0.1517854208  0.0437520021 
##            85            86            87            88            89 
##  0.4045665521 -0.2397558774 -0.0389659308  0.1668886675 -0.3382537203 
##            90            91            92            93            94 
## -0.5413580162 -0.1546854860  0.3520710326  0.4025921526  0.2095165604 
##            95            96            97            98            99 
##  0.0551532552  0.4117492477  0.2310340573  0.0619664336  0.3702852112 
##           100           101           102           103           104 
## -0.3659996520 -0.0044539364 -0.2820273933 -0.1702610548 -0.1867061572 
##           105           106           107           108           109 
##  0.0160634174 -0.0503426977  0.1004954227 -0.0847025062  0.0184641120 
##           110           111           112           113           114 
##  0.1677675811 -0.0045166105  0.1848077407  0.3155652851  0.0017123017 
##           115           116           117           118           119 
##  0.0201499198 -0.0072897578 -0.1471967533  0.0649892211 -0.2567447290 
##           120           121           122           123           124 
##  0.3180680699 -0.2578559732  0.1715460277 -0.0407654594  0.2050619264 
##           125           126           127           128           129 
## -0.1283455165  0.0464472788  0.1646379647 -0.0074208489  0.1114655544 
##           130           131           132           133           134 
##  0.1886244484 -0.1235962143  0.2089130225 -0.0777261807  0.0897534749 
##           135           136           137           138           139 
## -0.0392630622  0.3165360728 -0.1363644954  0.1981465552 -0.2215215698 
##           140           141           142           143           144 
##  0.2475332028  0.3910507079 -0.0781236710  0.1386567604 -0.1735720773 
##           145           146           147           148           149 
## -0.1667882723 -0.1602864360 -0.0017313771  0.2791168831 -1.2446589545 
##           150           151           152           153           154 
##  0.2010499865 -1.2316012038  0.1208550083  0.3942390124  0.3643330889 
##           155           156           157           158           159 
##  0.5994139528 -0.2251040535 -0.2553868236 -0.2098719206 -0.1423459045 
##           160           161           162           163           164 
## -0.6261824990  0.0746145589 -0.1036221899  0.3266902752  0.2285167608 
##           165           166           167           168           169 
##  0.1345033423  0.0234848398  0.3921030542 -0.1310248709 -0.2888057455 
##           170           171           172           173           174 
## -0.3346308722 -0.0178046585 -0.8237569096  0.0537727478 -0.2987336735 
##           175           176           177           178           179 
##  0.3920360799  0.0049644440  0.0364454745 -0.2400235480 -0.0319620907 
##           180           181           182           183           184 
##  0.0896111419  0.2158452519 -0.2042719776 -1.6466917035  0.5637835021 
##           185           186           187           188           189 
##  0.1385859502 -0.0084976133  0.1187905948  0.5661215724  0.1774999216 
##           190           191           192           193           194 
##  0.2929252839 -0.0722941997  0.0985391970 -0.3036262277  0.1865300795 
##           195           196           197           198           199 
## -0.3665561072 -0.0762739724  0.0135137090 -0.1920264988  0.3074981255 
##           200           201           202           203           204 
## -0.0088667829  0.3391955387  0.1045959119  0.2588369218  0.4291785045 
##           205           206           207           208           209 
##  0.0204098280  0.0184465190 -0.1104780230  0.2697034265  0.0028814696 
##           210           211           212           213           214 
##  0.0419644738 -0.0979494161  0.0868746075 -0.0970305611 -0.0670294633 
##           215           216           217           218           219 
##  0.1908512811 -0.1512890871  0.0255835992  0.1600611619  0.0034475921 
##           220           221           222           223           224 
##  0.0071120902 -0.0360415240  0.1586572222  0.0035606793 -0.0502878792 
##           225           226           227           228           229 
## -0.0403038784  0.1481848140 -0.0374356098  0.1107976213  0.1553037386 
##           230           231           232           233           234 
##  0.0477814486  0.1110411713 -0.1146757342  0.1969397528 -0.0027768938 
##           235           236           237           238           239 
##  0.1218550071  0.1514320783  0.1669940784  0.5912327499 -0.3766883900 
##           240           241           242           243           244 
## -0.0495979019 -0.1857916574  0.0988654429  0.2507321225  0.0135622288 
##           245           246           247           248           249 
##  0.0403873215 -0.5696748039 -1.5865873446  0.8919220254 -0.0822760186 
##           250           251           252           253           254 
## -0.0485748516  0.1630222222  0.3354818448 -0.2243762717  0.2646385836 
##           255           256           257           258           259 
## -0.4599378306  0.0119103222  0.1172958366 -0.0926178044 -0.0267898997 
##           260           261           262           263           264 
## -0.1741233441 -0.1423067638 -0.1683154951 -0.1076029647 -0.1993887590 
##           265           266           267           268           269 
##  0.0540048422 -0.2140946861 -0.4465152427  0.3935122069 -0.1229999148 
##           270           271           272           273           274 
## -0.0328911277  0.0831043035  0.0088594194  0.2488088994 -0.4089294928 
##           275           276           277           278           279 
## -0.1256825061 -0.2582741964  0.0213734176 -0.1355973731  0.7013449295 
##           280           281           282           283           284 
## -0.5002528665  0.4247785975 -0.2138842822  0.0249356409  0.4448587183 
##           285           286           287           288           289 
##  0.0732101229 -0.5124918716 -0.2490751165 -0.8424235208  0.2542447174 
##           290           291           292           293           294 
## -0.1568516733  0.1218880097 -2.4612987995 -0.3886303267  1.3165076657 
##           295           296           297           298           299 
## -0.1859296201  0.3581716204  0.6624890484  0.4770345997  0.2447427935 
##           300           301           302           303           304 
##  0.1344935681 -0.1787336647 -0.4601851144 -0.2134567950 -0.4630961972 
##           305           306           307           308           309 
## -0.1233729125 -0.4058441082 -0.0100965875 -0.0275912750  0.4892245503 
##           310           311           312           313           314 
## -0.2338331408 -0.9068582252  0.5513015642 -0.2406651326  0.4263839840 
##           315           316           317           318           319 
## -0.0683218294  0.2782753622  0.2140733302 -0.0657137151  0.1132415097 
##           320           321           322           323           324 
## -0.0196968837  0.0981828093  0.3842594035 -0.0561284384  0.0528987035 
##           325           326           327           328           329 
## -0.1754615429 -0.0856128581 -0.8145438611  0.1628236104  0.2825244641 
##           330           331           332           333           334 
## -0.0932301332  0.4279563417  0.2885995685  0.6202084727  0.1442063807 
##           335           336           337           338           339 
##  0.2764556537  0.0930344340  0.2241364101 -0.2362858700  0.0067862220 
##           340           341           342           343           344 
## -0.2526058306 -0.4311946803  0.1250792613 -0.0502285029  0.1786453660 
##           345           346           347           348           349 
## -0.0041684222 -0.1812127409 -0.0435386339  0.1040552939  0.1810831655 
##           350           351           352           353           354 
##  0.4964302549  0.2409475717  0.1230553936  0.2875661810  0.3296716572 
##           355           356           357           358           359 
##  0.5192556035 -0.2530760536 -0.5321046695  0.8648383778  0.8765670332 
##           360           361           362           363           364 
##  0.6149859799  0.4784652088  0.4985655865 -0.1843786068  0.0531959389 
##           365           366           367           368           369 
## -0.1285329564  0.1558907211  0.4319776920  0.1746592239  0.1331990611 
##           370           371           372           373           374 
##  0.1347729451 -0.0423535777 -0.5918717364  0.3549544233 -0.0123358534 
##           375           376           377           378           379 
##  0.1023419808  0.1924829564  0.1458843414 -0.1942461881 -1.2173498271 
##           380           381           382           383           384 
##  0.1361224701  0.5878242670  0.2082606231  0.1481623289 -0.8667974035 
##           385           386           387           388           389 
## -0.8770171340  0.4959182988  0.6233906734 -0.0277453856  0.1200462458 
##           390           391           392           393           394 
##  0.1846111916  0.0467849905 -0.6885086919 -0.1648419290  0.2642455003 
##           395           396           397           398           399 
## -0.0459592409 -0.1294998468 -0.0403567194  0.0717888017  0.1271250032 
##           400           401           402           403           404 
## -0.1045780612 -0.0738128574 -0.1548812858  0.0864681465 -0.3137185135 
##           405           406           407           408           409 
## -0.1977121611  0.0428431925  0.0588420687  0.0402211061 -0.0615905323 
##           410           411           412           413           414 
## -0.1365868529  0.1686192989 -0.1307640329 -0.2338499171  0.2827024019 
##           415           416           417           418           419 
## -0.1090935595 -0.2662020349 -0.2987347252 -0.1665490033 -0.0657298500 
##           420           421           422           423           424 
## -0.5255945312  0.0444977488 -0.2414322739 -0.2191448620  0.0930775880 
##           425           426           427           428           429 
##  0.1156247530 -0.7156550294 -0.0562302928  0.0976831542  0.1468853897 
##           430           431           432           433           434 
##  0.1722577857 -0.3585870807  0.4206892342 -0.0683656352 -0.0388879098 
##           435           436           437           438           439 
## -0.2062300059  0.1821954907 -0.3861866081  0.2023325465 -0.1072643478 
##           440           441           442           443           444 
## -0.1476424867 -0.1467677394  0.3493979779  0.0729184219 -0.3734969364 
##           445           446           447           448           449 
##  0.0175852067  0.3319980888 -0.0947587384 -0.3537263700 -0.1194274364 
##           450           451           452           453           454 
## -0.3513892994  0.2655532502 -0.3212382950 -0.0571953816  0.5528856841 
##           455           456           457           458           459 
## -0.1372914259  0.2515315767  0.0793066581 -0.2757035685  0.2815006917 
##           460           461           462           463           464 
## -0.0271778304  0.0318641099  0.2834100289 -0.0124619903 -0.3083795898 
##           465           466           467           468           469 
##  0.2075704614  0.2075792184 -0.0082125291 -0.1153333569  0.2389948463 
##           470           471           472           473           474 
## -0.1255877572 -0.0667826630  0.0521944024 -0.1502844440 -0.0495748138 
##           475           476           477           478           479 
##  0.0630449612 -0.0068500097  0.2853022030 -0.3005589208 -0.0057430159 
##           480           481           482           483           484 
##  0.0556608136 -0.1472646441  0.0758988594  0.0881283171 -0.2037462741 
##           485           486           487           488           489 
##  0.1289792378  0.0144332184  0.0072104087  0.2294987017 -0.0123383501 
##           490 
##  0.2074579792

mod_ardl92 <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                        data = full_cases_wastewater_weather_data_train, 
                        p=2,q=9)
summary(mod_ardl92 ) 
## 
## Time series regression with "ts" data:
## Start = 10, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.36421 -0.14943  0.01629  0.16652  1.45010 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.67598    0.23205  -2.913 0.003750 ** 
## log_viral_gene.t      0.02639    0.02498   1.056 0.291347    
## log_viral_gene.1      0.01316    0.03187   0.413 0.679773    
## log_viral_gene.2      0.01030    0.02521   0.409 0.683028    
## log_mean_new_cases.1  0.48306    0.04625  10.444  < 2e-16 ***
## log_mean_new_cases.2  0.12026    0.05128   2.345 0.019432 *  
## log_mean_new_cases.3  0.09138    0.05153   1.773 0.076823 .  
## log_mean_new_cases.4  0.10696    0.05167   2.070 0.038980 *  
## log_mean_new_cases.5  0.17515    0.05112   3.427 0.000665 ***
## log_mean_new_cases.6  0.02968    0.05163   0.575 0.565665    
## log_mean_new_cases.7  0.08478    0.05136   1.651 0.099470 .  
## log_mean_new_cases.8 -0.10149    0.05095  -1.992 0.046930 *  
## log_mean_new_cases.9 -0.08455    0.04554  -1.857 0.063994 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3343 on 468 degrees of freedom
## Multiple R-squared:  0.9119, Adjusted R-squared:  0.9096 
## F-statistic: 403.4 on 12 and 468 DF,  p-value: < 2.2e-16
f_ardl92 <- forecast(mod_ardl92  , 
                     x= t(full_cases_wastewater_weather_data_test[,7]),
                     h=14,
                     interval = TRUE)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl92$forecasts[,2])
## [1] 0.2526685
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
    f_ardl92$forecasts[,2])
## [1] 0.2293562
checkresiduals(mod_ardl92)
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -2.396407e-02 -3.976665e-02  2.253188e-02  1.126186e-02 -3.456715e-05 
##            15            16            17            18            19 
## -1.584383e-01 -1.087522e-01 -5.617950e-02 -2.259314e-02 -4.996504e-02 
##            20            21            22            23            24 
##  1.568234e-01  1.735783e-02 -2.082497e-01  2.457965e-01 -5.157538e-02 
##            25            26            27            28            29 
##  2.722065e-03 -2.621960e-01  2.818996e-01 -2.949834e-01 -3.035126e-01 
##            30            31            32            33            34 
##  1.661684e-01 -4.430982e-02  5.693436e-02  1.717116e-01  1.064284e-01 
##            35            36            37            38            39 
## -1.529803e-01 -2.945897e-01  1.263618e-01 -2.567127e-01 -1.473113e-02 
##            40            41            42            43            44 
##  5.214441e-02  2.746484e-02 -2.552454e-01 -8.854845e-02 -1.247769e-01 
##            45            46            47            48            49 
##  1.160506e-01  7.937280e-02 -6.261422e-02  9.856974e-02 -2.195059e-01 
##            50            51            52            53            54 
##  1.127137e-01 -1.766453e-01 -5.426369e-02 -4.002780e-01 -3.406671e-02 
##            55            56            57            58            59 
##  5.963782e-02  1.016042e-01  2.000816e-01  2.983519e-03 -2.650524e-01 
##            60            61            62            63            64 
##  1.486018e-01  1.601307e-01 -1.384878e-01  7.687082e-02 -4.141107e-02 
##            65            66            67            68            69 
##  1.098705e-01 -3.128669e-02  6.173563e-02  1.494617e-01  7.418673e-02 
##            70            71            72            73            74 
##  1.644328e-01  4.680422e-01  2.262063e-02 -1.274640e-01  5.006964e-02 
##            75            76            77            78            79 
## -3.068078e-01  8.782886e-02 -4.406198e-02 -2.869990e-03  5.181883e-03 
##            80            81            82            83            84 
##  5.072444e-02 -2.090606e-01  1.372967e-01 -1.820851e-01 -1.857852e-02 
##            85            86            87            88            89 
##  3.443533e-01 -2.852786e-01  2.303650e-02  2.023233e-01 -2.091241e-01 
##            90            91            92            93            94 
## -5.177255e-01 -1.381966e-01  3.400566e-01  3.701602e-01  2.276316e-01 
##            95            96            97            98            99 
##  1.177870e-01  3.214996e-01  1.667340e-01 -1.459042e-03  2.712643e-01 
##           100           101           102           103           104 
## -3.993105e-01 -1.274003e-02 -2.554181e-01 -1.053282e-01 -1.625378e-01 
##           105           106           107           108           109 
##  5.634557e-02 -2.966006e-02  1.075129e-01 -6.818631e-02  3.926711e-02 
##           110           111           112           113           114 
##  1.489118e-01 -5.768349e-02  1.701619e-01  2.698320e-01  4.012499e-03 
##           115           116           117           118           119 
##  6.400294e-03 -2.979161e-02 -1.944667e-01  5.016915e-02 -2.560521e-01 
##           120           121           122           123           124 
##  3.215355e-01 -2.638250e-01  1.665216e-01 -2.162241e-02  1.748811e-01 
##           125           126           127           128           129 
## -1.494302e-01  1.042969e-02  2.123463e-01 -2.841083e-03  1.362311e-01 
##           130           131           132           133           134 
##  1.325026e-01 -1.897168e-01  1.637814e-01 -3.943942e-02  1.827401e-02 
##           135           136           137           138           139 
## -5.139633e-02  2.554954e-01 -1.630690e-01  1.330048e-01 -2.957152e-01 
##           140           141           142           143           144 
##  1.841674e-01  2.507736e-01 -1.557800e-01  9.174971e-02 -2.204612e-01 
##           145           146           147           148           149 
## -1.119750e-01 -2.616129e-01 -6.771456e-02  1.873420e-01 -1.271189e+00 
##           150           151           152           153           154 
##  2.556409e-01 -1.170900e+00  1.055138e-01  4.032973e-01  3.828688e-01 
##           155           156           157           158           159 
##  5.568852e-01 -2.625216e-01 -3.221344e-01 -3.517223e-01 -2.326498e-01 
##           160           161           162           163           164 
## -6.413548e-01  1.490695e-01 -8.621020e-02  4.664573e-01  1.604656e-01 
##           165           166           167           168           169 
##  1.383717e-01 -2.395036e-02  3.262965e-01 -1.042147e-01 -3.213268e-01 
##           170           171           172           173           174 
## -3.663798e-01 -6.002995e-02 -7.778285e-01  9.547445e-02 -2.161891e-01 
##           175           176           177           178           179 
##  4.416636e-01  6.852906e-02  3.004097e-02 -2.704458e-01 -7.407742e-02 
##           180           181           182           183           184 
##  4.190433e-02  1.797658e-01 -2.287042e-01 -1.620557e+00  7.144359e-01 
##           185           186           187           188           189 
##  2.002879e-01  1.150878e-01  1.289638e-01  6.242632e-01  2.050394e-01 
##           190           191           192           193           194 
##  3.527073e-01 -1.933211e-02  7.994148e-02 -3.369943e-01  1.785146e-01 
##           195           196           197           198           199 
## -2.045247e-01  2.162620e-03  2.513039e-01 -2.552056e-02  3.232862e-01 
##           200           201           202           203           204 
## -6.735815e-02  3.296637e-01  8.652104e-02  2.380897e-01  4.655643e-01 
##           205           206           207           208           209 
##  8.662858e-02 -1.184034e-02 -9.723711e-02  2.729274e-01  8.313439e-03 
##           210           211           212           213           214 
##  1.015981e-01 -5.016132e-02  2.061002e-01 -5.190664e-02  1.400110e-02 
##           215           216           217           218           219 
##  2.209761e-01 -1.501411e-01  5.880174e-02  1.781954e-01  5.708528e-02 
##           220           221           222           223           224 
##  1.981210e-02  1.167935e-02  1.602544e-01  7.165049e-03 -3.680604e-02 
##           225           226           227           228           229 
## -5.092278e-03  1.754901e-01 -3.829182e-02  1.375325e-01  1.482769e-01 
##           230           231           232           233           234 
##  5.776840e-02  1.493228e-01 -9.650623e-02  2.228080e-01 -4.609739e-02 
##           235           236           237           238           239 
##  9.670582e-02  1.313394e-01  1.672115e-01  5.662791e-01 -3.761859e-01 
##           240           241           242           243           244 
## -6.367110e-02 -2.081374e-01  9.186764e-02  2.436832e-01  3.653342e-03 
##           245           246           247           248           249 
## -6.807823e-02 -6.105565e-01 -1.620877e+00  9.680602e-01  1.192657e-03 
##           250           251           252           253           254 
## -9.628339e-02  1.545319e-01  3.129851e-01 -1.631988e-01  2.493072e-01 
##           255           256           257           258           259 
## -4.769542e-01 -8.771902e-02 -2.167812e-03 -1.056218e-02  1.022641e-01 
##           260           261           262           263           264 
## -2.346255e-01  1.379507e-02 -2.023609e-01 -1.173748e-01 -2.047266e-01 
##           265           266           267           268           269 
##  6.049889e-02 -2.346834e-01 -4.229977e-01  3.988999e-01 -9.839163e-02 
##           270           271           272           273           274 
## -2.135352e-02  7.741357e-02  1.877729e-02  2.034479e-01 -4.268264e-01 
##           275           276           277           278           279 
## -1.360064e-01 -2.454923e-01  5.370585e-02 -6.652115e-02  7.476438e-01 
##           280           281           282           283           284 
## -5.191479e-01  4.549196e-01 -2.347329e-01  2.828300e-02  4.456982e-01 
##           285           286           287           288           289 
##  3.968121e-02 -4.771259e-01 -2.831081e-01 -7.737644e-01  3.006624e-01 
##           290           291           292           293           294 
## -1.023992e-01  1.395710e-01 -2.364213e+00 -3.909424e-01  1.450096e+00 
##           295           296           297           298           299 
## -2.085535e-01  2.979523e-01  6.068575e-01  3.951824e-01  2.037335e-01 
##           300           301           302           303           304 
##  8.261871e-02 -3.236322e-01 -5.990560e-01 -2.930634e-01 -2.168600e-01 
##           305           306           307           308           309 
## -4.583283e-02 -3.267313e-01  1.914202e-01 -1.029380e-01  4.824966e-01 
##           310           311           312           313           314 
## -2.567830e-01 -9.254083e-01  5.238682e-01 -2.883445e-01  4.218525e-01 
##           315           316           317           318           319 
## -1.152390e-01  2.896990e-01  2.318420e-01 -4.225929e-02  1.206934e-01 
##           320           321           322           323           324 
## -1.258189e-02  8.128629e-02  5.362193e-01  3.580403e-02  5.387543e-02 
##           325           326           327           328           329 
## -1.426912e-01 -2.952867e-01 -7.912501e-01  1.631587e-01  4.219926e-01 
##           330           331           332           333           334 
##  6.565432e-02  4.054452e-01  2.009018e-01  4.716507e-01  2.343397e-02 
##           335           336           337           338           339 
##  2.490811e-01  5.407757e-02  2.453663e-01 -2.296027e-01  3.777055e-02 
##           340           341           342           343           344 
## -2.720543e-01 -4.027320e-01  2.307273e-01 -3.164816e-02  2.747526e-01 
##           345           346           347           348           349 
## -4.497552e-03 -1.574489e-01 -4.906954e-02  8.772337e-02  1.370770e-01 
##           350           351           352           353           354 
##  4.529530e-01  1.906016e-01  1.525708e-01  3.284963e-01  3.650998e-01 
##           355           356           357           358           359 
##  5.396324e-01 -3.074325e-01 -5.349434e-01  8.777813e-01  9.443478e-01 
##           360           361           362           363           364 
##  7.003203e-01  4.952868e-01  5.047689e-01 -1.918388e-01  5.261175e-02 
##           365           366           367           368           369 
## -1.077546e-01  1.598769e-01  4.668144e-01  3.266047e-01  2.771328e-01 
##           370           371           372           373           374 
##  1.589788e-01  1.313453e-02 -6.308058e-01  3.184335e-01 -6.444854e-03 
##           375           376           377           378           379 
##  1.939464e-01  1.715339e-01  1.996648e-01 -3.091449e-01 -1.296596e+00 
##           380           381           382           383           384 
##  1.088063e-01  5.960841e-01  2.713044e-01  2.329620e-01 -8.556874e-01 
##           385           386           387           388           389 
## -9.022326e-01  4.009147e-01  5.146857e-01 -9.337388e-02  1.724957e-01 
##           390           391           392           393           394 
##  1.810539e-01  1.784066e-01 -8.606756e-01 -2.221742e-01  1.713428e-01 
##           395           396           397           398           399 
## -1.589677e-01 -2.242434e-03  6.841561e-02  4.039676e-02  1.309074e-01 
##           400           401           402           403           404 
## -1.937431e-01 -1.420454e-01 -2.128705e-01  1.629227e-02 -2.543536e-01 
##           405           406           407           408           409 
## -2.542008e-01  5.524668e-03  3.713062e-02 -3.802828e-02 -8.559204e-02 
##           410           411           412           413           414 
## -1.088547e-01  9.989792e-02 -1.654432e-01 -3.312310e-01  2.208590e-01 
##           415           416           417           418           419 
## -1.385087e-01 -2.478310e-01 -2.713380e-01 -1.457082e-01 -6.735750e-02 
##           420           421           422           423           424 
## -5.499413e-01  1.577297e-02 -2.629274e-01 -2.215989e-01  1.088316e-01 
##           425           426           427           428           429 
##  1.518798e-01 -7.800507e-01  4.120184e-02  8.104201e-02  1.116859e-01 
##           430           431           432           433           434 
##  6.589971e-02 -5.915863e-01  3.969890e-01 -1.167412e-01  5.200500e-02 
##           435           436           437           438           439 
## -1.569589e-01  6.584974e-02 -5.131337e-01  1.382272e-01 -1.724006e-01 
##           440           441           442           443           444 
## -9.139437e-02 -9.756700e-02  4.205117e-01  9.049092e-02 -4.838474e-01 
##           445           446           447           448           449 
##  7.527956e-03  3.232683e-01 -1.284454e-01 -3.784562e-01 -9.697398e-02 
##           450           451           452           453           454 
## -3.759760e-01  3.639162e-01 -2.447045e-01 -3.919063e-02  5.256440e-01 
##           455           456           457           458           459 
## -1.279611e-01  2.734360e-01  6.062632e-02 -2.399165e-01  3.003080e-01 
##           460           461           462           463           464 
## -4.265930e-02  5.641235e-02  3.142238e-01  2.458586e-02 -1.890712e-01 
##           465           466           467           468           469 
##  2.477738e-01  2.543826e-01  8.183976e-02 -1.401468e-01  3.270317e-01 
##           470           471           472           473           474 
## -8.675708e-02 -2.404920e-02  1.507883e-01 -1.698617e-01 -3.771766e-02 
##           475           476           477           478           479 
##  9.785553e-02  7.054171e-02  3.302712e-01 -2.662845e-01  8.830143e-03 
##           480           481           482           483           484 
##  2.716887e-03 -1.649914e-01  1.160813e-01  1.064102e-01 -1.588147e-01 
##           485           486           487           488           489 
##  1.602108e-01  3.967555e-02  3.183249e-02  2.427844e-01 -4.431468e-03 
##           490 
##  2.543166e-01

exp(f_ardl92$forecasts[1,2])
## [1] 5.333528
exp(f_ardl92$forecasts[1,1])
## [1] 2.934801
exp(f_ardl92$forecasts[1,3])
## [1] 9.972517
exp(f_ardl92$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_test[1,6])
## [1] -3.384016
exp(f_ardl92$forecasts[7,2])
## [1] 5.891802
exp(f_ardl92$forecasts[7,1])
## [1] 2.248932
exp(f_ardl92$forecasts[7,3])
## [1] 14.95402
exp(f_ardl92$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_test[7,6])
## [1] -1.10645
exp(f_ardl92$forecasts[14,2])
## [1] 6.198724
exp(f_ardl92$forecasts[14,1])
## [1] 2.024789
exp(f_ardl92$forecasts[14,3])
## [1] 19.50234
exp(f_ardl92$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_test[14,6])
## [1] 0.4920041
lowest_rmse_weather <- Inf
best_mod_weather <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    remove <- list(p =list(TAVG=c(1:p),mean_precipation=c(1:p)))
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                     TAVG,data = full_cases_wastewater_weather_data_train,
                   p=p,q=q,
                   remove = remove)
    f <- forecast(mod, x= t(full_cases_wastewater_weather_data_test[,c(7,4,5)]),h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
                         f$forecasts) #interchanged between RMSE and MAE 
    if (forecast_acc<lowest_rmse_weather){
      lowest_rmse_weather <- forecast_acc
      best_mod_weather <- mod 
    }
  }
}

lowest_rmse_weather #0.20
## [1] 0.2001624
summary(best_mod_weather) #ARDL(13,11)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.43826 -0.14473  0.01622  0.16796  1.37242 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -1.0012439  0.3182374  -3.146 0.001764 ** 
## log_viral_gene.t       0.0219351  0.0264055   0.831 0.406583    
## log_viral_gene.1       0.0174653  0.0330155   0.529 0.597066    
## log_viral_gene.2      -0.0198144  0.0331529  -0.598 0.550363    
## log_viral_gene.3       0.0257640  0.0332502   0.775 0.438834    
## log_viral_gene.4      -0.0006246  0.0332382  -0.019 0.985016    
## log_viral_gene.5       0.0271930  0.0331620   0.820 0.412648    
## log_viral_gene.6       0.0135525  0.0331998   0.408 0.683313    
## log_viral_gene.7      -0.0291835  0.0332550  -0.878 0.380649    
## log_viral_gene.8      -0.0072037  0.0331655  -0.217 0.828149    
## log_viral_gene.9       0.0132099  0.0328965   0.402 0.688200    
## log_viral_gene.10     -0.0059881  0.0327731  -0.183 0.855105    
## log_viral_gene.11      0.0133795  0.0262905   0.509 0.611065    
## mean_precipation.t    -0.0456209  0.0514223  -0.887 0.375456    
## TAVG.t                 0.0009141  0.0012119   0.754 0.451100    
## log_mean_new_cases.1   0.4663508  0.0472625   9.867  < 2e-16 ***
## log_mean_new_cases.2   0.1182938  0.0522279   2.265 0.023991 *  
## log_mean_new_cases.3   0.0779376  0.0526102   1.481 0.139197    
## log_mean_new_cases.4   0.1024029  0.0529550   1.934 0.053769 .  
## log_mean_new_cases.5   0.1838986  0.0531158   3.462 0.000587 ***
## log_mean_new_cases.6   0.0494224  0.0538579   0.918 0.359298    
## log_mean_new_cases.7   0.1092176  0.0534975   2.042 0.041781 *  
## log_mean_new_cases.8  -0.0885420  0.0536626  -1.650 0.099647 .  
## log_mean_new_cases.9  -0.0536585  0.0530340  -1.012 0.312190    
## log_mean_new_cases.10  0.0265351  0.0529905   0.501 0.616791    
## log_mean_new_cases.11 -0.0215972  0.0528675  -0.409 0.683090    
## log_mean_new_cases.12 -0.0910823  0.0524895  -1.735 0.083383 .  
## log_mean_new_cases.13 -0.0010250  0.0475941  -0.022 0.982827    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.336 on 449 degrees of freedom
## Multiple R-squared:  0.9139, Adjusted R-squared:  0.9087 
## F-statistic: 176.5 on 27 and 449 DF,  p-value: < 2.2e-16
remove <- list(p =list(TAVG=c(1:11),mean_precipation=c(1:11)))
mod_ardl1311_weather <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                 TAVG,data = full_cases_wastewater_weather_data_train, 
               p=11,q=13,
               remove = remove)
summary(mod_ardl1311_weather)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.43826 -0.14473  0.01622  0.16796  1.37242 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -1.0012439  0.3182374  -3.146 0.001764 ** 
## log_viral_gene.t       0.0219351  0.0264055   0.831 0.406583    
## log_viral_gene.1       0.0174653  0.0330155   0.529 0.597066    
## log_viral_gene.2      -0.0198144  0.0331529  -0.598 0.550363    
## log_viral_gene.3       0.0257640  0.0332502   0.775 0.438834    
## log_viral_gene.4      -0.0006246  0.0332382  -0.019 0.985016    
## log_viral_gene.5       0.0271930  0.0331620   0.820 0.412648    
## log_viral_gene.6       0.0135525  0.0331998   0.408 0.683313    
## log_viral_gene.7      -0.0291835  0.0332550  -0.878 0.380649    
## log_viral_gene.8      -0.0072037  0.0331655  -0.217 0.828149    
## log_viral_gene.9       0.0132099  0.0328965   0.402 0.688200    
## log_viral_gene.10     -0.0059881  0.0327731  -0.183 0.855105    
## log_viral_gene.11      0.0133795  0.0262905   0.509 0.611065    
## mean_precipation.t    -0.0456209  0.0514223  -0.887 0.375456    
## TAVG.t                 0.0009141  0.0012119   0.754 0.451100    
## log_mean_new_cases.1   0.4663508  0.0472625   9.867  < 2e-16 ***
## log_mean_new_cases.2   0.1182938  0.0522279   2.265 0.023991 *  
## log_mean_new_cases.3   0.0779376  0.0526102   1.481 0.139197    
## log_mean_new_cases.4   0.1024029  0.0529550   1.934 0.053769 .  
## log_mean_new_cases.5   0.1838986  0.0531158   3.462 0.000587 ***
## log_mean_new_cases.6   0.0494224  0.0538579   0.918 0.359298    
## log_mean_new_cases.7   0.1092176  0.0534975   2.042 0.041781 *  
## log_mean_new_cases.8  -0.0885420  0.0536626  -1.650 0.099647 .  
## log_mean_new_cases.9  -0.0536585  0.0530340  -1.012 0.312190    
## log_mean_new_cases.10  0.0265351  0.0529905   0.501 0.616791    
## log_mean_new_cases.11 -0.0215972  0.0528675  -0.409 0.683090    
## log_mean_new_cases.12 -0.0910823  0.0524895  -1.735 0.083383 .  
## log_mean_new_cases.13 -0.0010250  0.0475941  -0.022 0.982827    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.336 on 449 degrees of freedom
## Multiple R-squared:  0.9139, Adjusted R-squared:  0.9087 
## F-statistic: 176.5 on 27 and 449 DF,  p-value: < 2.2e-16
f_ardl1311_weather <- forecast(mod_ardl1311_weather, 
                               x= t(full_cases_wastewater_weather_data_test[,c(7,4,5)]),
                               h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl1311_weather$forecasts) 
## [1] 0.2001624
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl1311_weather$forecasts) 
## [1] 0.1641279
checkresiduals(mod_ardl1311_weather)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##           14           15           16           17           18           19 
##  0.032061005 -0.150376998 -0.096764266 -0.024326879 -0.006033189 -0.061639427 
##           20           21           22           23           24           25 
##  0.172313111  0.036109140 -0.163153524  0.306839066 -0.023248073  0.042988574 
##           26           27           28           29           30           31 
## -0.228792606  0.291939369 -0.253214936 -0.286974412  0.157805177 -0.041882791 
##           32           33           34           35           36           37 
##  0.092751873  0.248168218  0.120988283 -0.075388784 -0.249685960  0.119038905 
##           38           39           40           41           42           43 
## -0.226425300  0.024053522  0.092684854  0.049254789 -0.197106402 -0.031675197 
##           44           45           46           47           48           49 
## -0.085013388  0.154805313  0.122142247 -0.032640348  0.097755610 -0.173631705 
##           50           51           52           53           54           55 
##  0.102965451 -0.140866728 -0.058196482 -0.403195095 -0.033118074  0.064499978 
##           56           57           58           59           60           61 
##  0.083693388  0.195087344  0.022111441 -0.245788240  0.172196035  0.130329062 
##           62           63           64           65           66           67 
## -0.107403410  0.096496264  0.033249704  0.145769693 -0.031483364  0.035262781 
##           68           69           70           71           72           73 
##  0.210939698  0.122008498  0.222762916  0.486282374  0.054789733 -0.061220550 
##           74           75           76           77           78           79 
##  0.068447820 -0.342555827  0.109303316 -0.057485398  0.074907367  0.047659920 
##           80           81           82           83           84           85 
##  0.049316357 -0.303508562  0.049874857 -0.152739977  0.069444011  0.385117934 
##           86           87           88           89           90           91 
## -0.252648087 -0.040845346  0.186709665 -0.289181694 -0.526622267 -0.154233204 
##           92           93           94           95           96           97 
##  0.355637101  0.400873761  0.198219842  0.065818752  0.396338656  0.228953050 
##           98           99          100          101          102          103 
##  0.058659003  0.312909049 -0.382026222 -0.021021910 -0.312984839 -0.177907468 
##          104          105          106          107          108          109 
## -0.160959236  0.049734977 -0.034218415  0.100279218 -0.080775539  0.041343224 
##          110          111          112          113          114          115 
##  0.175817973 -0.007346746  0.196050686  0.275538042  0.003330340 -0.009581105 
##          116          117          118          119          120          121 
## -0.020108093 -0.164347421  0.059155062 -0.265743595  0.301212802 -0.269953292 
##          122          123          124          125          126          127 
##  0.172284520 -0.027231693  0.208312479 -0.103518220  0.043189146  0.163847678 
##          128          129          130          131          132          133 
##  0.003105353  0.116445829  0.191607376 -0.103180847  0.202563358 -0.043867650 
##          134          135          136          137          138          139 
##  0.059618117 -0.036371240  0.305996882 -0.139523286  0.167535828 -0.232367932 
##          140          141          142          143          144          145 
##  0.231326487  0.357349545 -0.085443014  0.111011907 -0.206700103 -0.175970280 
##          146          147          148          149          150          151 
## -0.190339179 -0.007742491  0.268882485 -1.239673123  0.177793074 -1.154297496 
##          152          153          154          155          156          157 
##  0.131496350  0.403115037  0.358667026  0.574829088 -0.240887347 -0.279459683 
##          158          159          160          161          162          163 
## -0.217093162 -0.128183895 -0.636019502  0.051683543 -0.180209350  0.372307354 
##          164          165          166          167          168          169 
##  0.179834935  0.196040077  0.050758156  0.369100885 -0.133794298 -0.314467700 
##          170          171          172          173          174          175 
## -0.350953918 -0.007907322 -0.809125155  0.040713872 -0.261818603  0.401976753 
##          176          177          178          179          180          181 
##  0.022231672  0.022028720 -0.253000288 -0.057681627  0.112242856  0.213332470 
##          182          183          184          185          186          187 
## -0.236490610 -1.666117004  0.565193241  0.121877827  0.022131507  0.224330207 
##          188          189          190          191          192          193 
##  0.617457991  0.183178578  0.274365696 -0.082064447  0.086382726 -0.303353346 
##          194          195          196          197          198          199 
##  0.172335233 -0.402559087 -0.124805257  0.156135222 -0.159500725  0.309056611 
##          200          201          202          203          204          205 
## -0.016480849  0.357921288  0.083388823  0.245893505  0.411978780  0.010134125 
##          206          207          208          209          210          211 
## -0.023043297 -0.111541860  0.239816911 -0.011807657  0.034113355 -0.112054582 
##          212          213          214          215          216          217 
##  0.106882353 -0.096712172 -0.055318704  0.183312623 -0.140600317  0.043283972 
##          218          219          220          221          222          223 
##  0.126125438 -0.015719030 -0.015973782 -0.041510343  0.129272034 -0.022913503 
##          224          225          226          227          228          229 
## -0.045754274 -0.030335738  0.133137226 -0.045719894  0.093117340  0.138695226 
##          230          231          232          233          234          235 
##  0.051993908  0.098095707 -0.128932499  0.167959783 -0.030371028  0.107488193 
##          236          237          238          239          240          241 
##  0.115530091  0.136336309  0.565503349 -0.401424554 -0.090609426 -0.205661527 
##          242          243          244          245          246          247 
##  0.087426181  0.221670423 -0.019028050  0.001623568 -0.592362525 -1.614036292 
##          248          249          250          251          252          253 
##  0.868355416 -0.087685692 -0.019022951  0.129980643  0.294339087 -0.219017717 
##          254          255          256          257          258          259 
##  0.221685049 -0.480412584 -0.031807385  0.069297095 -0.136699218 -0.089956797 
##          260          261          262          263          264          265 
## -0.227832079 -0.032101635 -0.193758911 -0.107770820 -0.219255032  0.036510746 
##          266          267          268          269          270          271 
## -0.238887761 -0.452834250  0.359906923 -0.128241075 -0.054474895  0.060718711 
##          272          273          274          275          276          277 
## -0.003573896  0.206321888 -0.431679482 -0.147878048 -0.278433330  0.004269636 
##          278          279          280          281          282          283 
## -0.145240305  0.795038513 -0.483146613  0.445424917 -0.233729439  0.013174370 
##          284          285          286          287          288          289 
##  0.426288670  0.046323401 -0.528314549 -0.284140202 -0.815833504  0.247695217 
##          290          291          292          293          294          295 
## -0.144209513  0.118142450 -2.438256210 -0.433856284  1.372419693 -0.210452936 
##          296          297          298          299          300          301 
##  0.385413737  0.673003665  0.488546755  0.227292166  0.101759843 -0.230520835 
##          302          303          304          305          306          307 
## -0.450881823 -0.243036654 -0.478977424 -0.218770399 -0.285253927  0.124793298 
##          308          309          310          311          312          313 
## -0.060504467  0.520401723 -0.226305752 -0.918235706  0.532619990 -0.248045754 
##          314          315          316          317          318          319 
##  0.399323330 -0.108377421  0.297233428  0.238884298 -0.061252332  0.093941663 
##          320          321          322          323          324          325 
## -0.002586508  0.120190993  0.419249838 -0.054135049  0.062405319 -0.088480274 
##          326          327          328          329          330          331 
## -0.084194447 -0.772760109  0.169526469  0.322643827 -0.075520003  0.422057617 
##          332          333          334          335          336          337 
##  0.304822179  0.618618225  0.137906359  0.283365138  0.059030578  0.211968946 
##          338          339          340          341          342          343 
## -0.252468335  0.011854684 -0.241413399 -0.370068490  0.151858385 -0.036801846 
##          344          345          346          347          348          349 
##  0.212708386  0.006164133 -0.145215431 -0.050753840  0.109360735  0.156216692 
##          350          351          352          353          354          355 
##  0.488398165  0.258612563  0.144476868  0.303299240  0.335362105  0.551202815 
##          356          357          358          359          360          361 
## -0.281502103 -0.538542521  0.849285686  0.877310067  0.611771106  0.481068300 
##          362          363          364          365          366          367 
##  0.515190224 -0.170546097  0.039740992 -0.030973905  0.190796788  0.453644618 
##          368          369          370          371          372          373 
##  0.200631752  0.121508693  0.171099341  0.054210509 -0.548191005  0.355961935 
##          374          375          376          377          378          379 
## -0.014790161  0.097820870  0.160768848  0.163398034 -0.166445964 -1.167861458 
##          380          381          382          383          384          385 
##  0.138438494  0.575983442  0.188435239  0.152091778 -0.845138907 -0.877591360 
##          386          387          388          389          390          391 
##  0.527000774  0.598915649 -0.049380973  0.110199450  0.199436261  0.019739784 
##          392          393          394          395          396          397 
## -0.735478112 -0.119181883  0.276424243 -0.072481730 -0.189465299 -0.110170384 
##          398          399          400          401          402          403 
##  0.093350221  0.207852007 -0.072879502 -0.090193073 -0.176780010  0.065467848 
##          404          405          406          407          408          409 
## -0.339613881 -0.255417330  0.065369628  0.072541504  0.016215155 -0.072539378 
##          410          411          412          413          414          415 
## -0.154432700  0.159027761 -0.148005904 -0.260272081  0.266219102 -0.138049227 
##          416          417          418          419          420          421 
## -0.279057437 -0.322112511 -0.160930245 -0.054283488 -0.527094556  0.047840976 
##          422          423          424          425          426          427 
## -0.245042592 -0.221495954  0.086591334  0.102312857 -0.714592019 -0.056795039 
##          428          429          430          431          432          433 
##  0.082415760  0.125096134  0.159796788 -0.349543148  0.417316305 -0.028228935 
##          434          435          436          437          438          439 
##  0.005114253 -0.234829321  0.163411928 -0.392724810  0.237198533 -0.144727289 
##          440          441          442          443          444          445 
## -0.127237172 -0.134152750  0.363083941  0.066327482 -0.419665994  0.069049349 
##          446          447          448          449          450          451 
##  0.327717598 -0.078593109 -0.376452499 -0.112873125 -0.345181157  0.294103245 
##          452          453          454          455          456          457 
## -0.331954764 -0.027661647  0.567713281 -0.106760640  0.243815109  0.045540692 
##          458          459          460          461          462          463 
## -0.229800735  0.281965947 -0.033474850  0.020419616  0.296290952 -0.007220395 
##          464          465          466          467          468          469 
## -0.282857319  0.183976095  0.217966974  0.013014735 -0.141571276  0.259893790 
##          470          471          472          473          474          475 
## -0.102575592 -0.006328713  0.090598793 -0.155632379 -0.049319827  0.066040565 
##          476          477          478          479          480          481 
## -0.023419397  0.270277139 -0.293031679 -0.002861828  0.043219488 -0.164362541 
##          482          483          484          485          486          487 
##  0.086508499  0.057996589 -0.197018256  0.116034046 -0.015890435 -0.002435516 
##          488          489          490 
##  0.216266189 -0.001900728  0.238943022

remove <- list(p =list(TAVG=c(1:13),mean_precipation=c(1:13)))
mod_ardl813_weather <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                  TAVG,data = full_cases_wastewater_weather_data_train, 
                                p=13,q=8,
                                remove = remove)
summary(mod_ardl813_weather)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.39428 -0.14189  0.01805  0.15651  1.37259 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.780751   0.328849  -2.374  0.01800 *  
## log_viral_gene.t      0.031172   0.026114   1.194  0.23322    
## log_viral_gene.1      0.013489   0.033039   0.408  0.68327    
## log_viral_gene.2     -0.022961   0.033116  -0.693  0.48844    
## log_viral_gene.3      0.027153   0.033128   0.820  0.41286    
## log_viral_gene.4     -0.006753   0.033093  -0.204  0.83841    
## log_viral_gene.5      0.035553   0.033149   1.073  0.28406    
## log_viral_gene.6      0.019048   0.033086   0.576  0.56509    
## log_viral_gene.7     -0.034225   0.033052  -1.035  0.30099    
## log_viral_gene.8     -0.012107   0.033056  -0.366  0.71433    
## log_viral_gene.9      0.011679   0.032964   0.354  0.72328    
## log_viral_gene.10    -0.005044   0.032838  -0.154  0.87799    
## log_viral_gene.11     0.051869   0.032833   1.580  0.11485    
## log_viral_gene.12    -0.060364   0.032828  -1.839  0.06660 .  
## log_viral_gene.13     0.003535   0.026076   0.136  0.89223    
## mean_precipation.t   -0.042300   0.051342  -0.824  0.41044    
## TAVG.t                0.001220   0.001205   1.012  0.31221    
## log_mean_new_cases.1  0.486144   0.046723  10.405  < 2e-16 ***
## log_mean_new_cases.2  0.114132   0.051859   2.201  0.02825 *  
## log_mean_new_cases.3  0.090453   0.052136   1.735  0.08343 .  
## log_mean_new_cases.4  0.093774   0.052026   1.802  0.07214 .  
## log_mean_new_cases.5  0.154870   0.052092   2.973  0.00311 ** 
## log_mean_new_cases.6  0.027813   0.052734   0.527  0.59816    
## log_mean_new_cases.7  0.081100   0.052305   1.551  0.12172    
## log_mean_new_cases.8 -0.138042   0.047154  -2.927  0.00359 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3363 on 452 degrees of freedom
## Multiple R-squared:  0.9132, Adjusted R-squared:  0.9086 
## F-statistic: 198.1 on 24 and 452 DF,  p-value: < 2.2e-16
f_ardl813_weather <- forecast(mod_ardl813_weather, x= t(full_cases_wastewater_weather_data_test[,c(7,4,5)]),h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl813_weather$forecasts) 
## [1] 0.2383604
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
    f_ardl813_weather$forecasts) 
## [1] 0.2170364
checkresiduals(mod_ardl813_weather)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
## -0.0051873382 -0.1653064585 -0.1163634279 -0.0375845687 -0.0175413444 
##            19            20            21            22            23 
## -0.0683182714  0.1664994410 -0.0041764018 -0.1751808550  0.2887896037 
##            24            25            26            27            28 
## -0.0346758580  0.0278215728 -0.2422329156  0.3188187442 -0.2184220745 
##            29            30            31            32            33 
## -0.2914196372  0.1528080588 -0.0441128024  0.0797702218  0.2521223999 
##            34            35            36            37            38 
##  0.1218841956 -0.0996323184 -0.2996064341  0.0964367171 -0.1542556035 
##            39            40            41            42            43 
## -0.0295945517  0.0943436925  0.0637803014 -0.2604055595 -0.0472540796 
##            44            45            46            47            48 
## -0.1144520256  0.1575733660  0.0813929507 -0.0276815025  0.1271647948 
##            49            50            51            52            53 
## -0.1983802153  0.1105048981 -0.1373843596 -0.0735675271 -0.3803912194 
##            54            55            56            57            58 
##  0.0124732328  0.1094020412  0.1184034964  0.1713610778  0.0104734599 
##            59            60            61            62            63 
## -0.2214916294  0.1565147162  0.1179143538 -0.0920402990  0.0848260284 
##            64            65            66            67            68 
##  0.0447030855  0.1974672353 -0.0205232240  0.0104411066  0.1998789744 
##            69            70            71            72            73 
##  0.0836320513  0.2652872617  0.4583685027 -0.0196192134 -0.1064039827 
##            74            75            76            77            78 
##  0.0457793081 -0.3509960448  0.1228418338 -0.1280447176  0.1375123067 
##            79            80            81            82            83 
##  0.0853469572  0.0342799051 -0.3064300906  0.0172416000 -0.2133352522 
##            84            85            86            87            88 
##  0.1267071847  0.3544217167 -0.2814405836 -0.1178328122  0.2101637301 
##            89            90            91            92            93 
## -0.2762129034 -0.4700468485 -0.1268623458  0.4059115385  0.4646675521 
##            94            95            96            97            98 
##  0.1528528079  0.0787409754  0.3706523570  0.1679711186  0.1356300107 
##            99           100           101           102           103 
##  0.2726857456 -0.4320266015 -0.0133018731 -0.2769595940 -0.1346419615 
##           104           105           106           107           108 
## -0.1413227260 -0.0307865261 -0.0126902319  0.1163825686 -0.1367884398 
##           109           110           111           112           113 
##  0.0425386041  0.1371269310 -0.0304701045  0.2029752313  0.2608837757 
##           114           115           116           117           118 
## -0.0081808528 -0.0218942909 -0.0295312131 -0.1577842448  0.0830081751 
##           119           120           121           122           123 
## -0.2670651442  0.3227703615 -0.2853897380  0.1882122677 -0.0593486445 
##           124           125           126           127           128 
##  0.1620758755 -0.1485627376  0.0309397493  0.1334040618  0.0105754533 
##           129           130           131           132           133 
##  0.0737371692  0.2154714639 -0.0803973775  0.2097987706 -0.0578103353 
##           134           135           136           137           138 
##  0.0559168700 -0.0453211071  0.3694118359 -0.1954693572  0.1094317289 
##           139           140           141           142           143 
## -0.3253266427  0.1748081955  0.2715257799 -0.1636542357  0.0730967933 
##           144           145           146           147           148 
## -0.2895545435 -0.2636586826 -0.2263794305 -0.0690955565  0.2579213860 
##           149           150           151           152           153 
## -1.2736613389  0.0804444096 -1.1575452461  0.1270645357  0.3821784480 
##           154           155           156           157           158 
##  0.3631252459  0.4441244745 -0.3328207957 -0.3979061914 -0.1798751643 
##           159           160           161           162           163 
## -0.2206018779 -0.5771425318  0.1447477529 -0.1751909778  0.5267562612 
##           164           165           166           167           168 
##  0.1276995551  0.1667923852 -0.0325130820  0.2416780742 -0.1388076396 
##           169           170           171           172           173 
## -0.2233003975 -0.3601108477  0.0540228086 -0.7453759445  0.0857702308 
##           174           175           176           177           178 
## -0.2841412809  0.4314751558  0.0547203822 -0.0429921530 -0.3068464038 
##           179           180           181           182           183 
## -0.1226693674  0.0480735195  0.1957015810 -0.2116218080 -1.6112976360 
##           184           185           186           187           188 
##  0.6314572207  0.1576141778  0.0910491580  0.2241770584  0.6370442518 
##           189           190           191           192           193 
##  0.1600776551  0.2977000002 -0.1025107497  0.1906710796 -0.3091179052 
##           194           195           196           197           198 
##  0.3091661006 -0.1645442821 -0.0147840824  0.2213724219 -0.1177088281 
##           199           200           201           202           203 
##  0.4038341512 -0.0141317949  0.3579796362  0.0962171505  0.1406160888 
##           204           205           206           207           208 
##  0.4352171406  0.1041273442 -0.0676165610 -0.0837100932  0.2463939436 
##           209           210           211           212           213 
##  0.0302954282 -0.0274926655 -0.0502113374  0.1737021082 -0.0837943138 
##           214           215           216           217           218 
## -0.0077268787  0.2051408619 -0.1237119351  0.0180527670  0.1634375672 
##           219           220           221           222           223 
##  0.0116094141 -0.0127999991 -0.0527097293  0.1404063676  0.0037636160 
##           224           225           226           227           228 
## -0.0719122476 -0.0004504518  0.1299758026 -0.0591641618  0.1153344614 
##           229           230           231           232           233 
##  0.1539395905  0.0641532053  0.0686149730 -0.1229168771  0.1891402473 
##           234           235           236           237           238 
## -0.0501558667  0.1086664743  0.1297044575  0.1351858713  0.5180724583 
##           239           240           241           242           243 
## -0.4126949586 -0.0688489444 -0.2372214493  0.0639085113  0.1973365632 
##           244           245           246           247           248 
## -0.0284356439 -0.0665987015 -0.5988653109 -1.6575008162  0.8801420982 
##           249           250           251           252           253 
## -0.1664375431 -0.0677627368  0.1332383476  0.2497281305 -0.2711441351 
##           254           255           256           257           258 
##  0.0090925112 -0.5570103557  0.0636871578  0.0336488116 -0.0272459602 
##           259           260           261           262           263 
##  0.0700034391 -0.2521988773 -0.0442511905 -0.1001731092 -0.1449781304 
##           264           265           266           267           268 
## -0.2163418412  0.0479666012 -0.2771770402 -0.4304125595  0.3408008863 
##           269           270           271           272           273 
## -0.1268765048 -0.0694304445  0.0363978428 -0.0051007387  0.1941851140 
##           274           275           276           277           278 
## -0.4456793594 -0.1418878726 -0.2272175311 -0.0014621399 -0.0969365037 
##           279           280           281           282           283 
##  0.8537478644 -0.4983292141  0.4638242742 -0.2822070815  0.0384115307 
##           284           285           286           287           288 
##  0.4212965207  0.0396252839 -0.4741300867 -0.2119556397 -0.8172032139 
##           289           290           291           292           293 
##  0.3417350363 -0.1402497508  0.1233461437 -2.3942839623 -0.4291404738 
##           294           295           296           297           298 
##  1.3725893942 -0.2209286467  0.2973995088  0.6116357370  0.3931781847 
##           299           300           301           302           303 
##  0.1764194831  0.0037992087 -0.1970102819 -0.4310367426 -0.2506721859 
##           304           305           306           307           308 
## -0.2010376400 -0.0677549995 -0.3072746859  0.1312025451  0.0223471858 
##           309           310           311           312           313 
##  0.4418282535 -0.2899685251 -0.9459852229  0.4919761949 -0.2944838206 
##           314           315           316           317           318 
##  0.4509392582 -0.0845006019  0.2861663766  0.2177041939 -0.0813532082 
##           319           320           321           322           323 
##  0.0853353293  0.0938177747  0.1085938427  0.5098655904  0.0421073464 
##           324           325           326           327           328 
##  0.0793070934 -0.0346131007 -0.0461671901 -0.6742921356  0.2408128968 
##           329           330           331           332           333 
##  0.3372120287 -0.0423324737  0.4621978250  0.2150771126  0.5376169379 
##           334           335           336           337           338 
##  0.0213225229  0.0632211617  0.0570049633  0.2374807442 -0.1574028638 
##           339           340           341           342           343 
##  0.1068221213 -0.1940221302 -0.3340268130  0.1238332805  0.0140028059 
##           344           345           346           347           348 
##  0.2381280215  0.0324723967 -0.1739220327 -0.0736422419  0.0483268164 
##           349           350           351           352           353 
##  0.0887470767  0.4918377839  0.2713118197  0.1484605854  0.3183614243 
##           354           355           356           357           358 
##  0.3250672275  0.5693584962 -0.2743996166 -0.4988077412  0.9445885236 
##           359           360           361           362           363 
##  0.8930693282  0.6844307827  0.5089700001  0.4997961140 -0.1204970646 
##           364           365           366           367           368 
##  0.0337010488  0.0094604438  0.2829888281  0.5056383526  0.2962405712 
##           369           370           371           372           373 
##  0.2322465366  0.1538246916 -0.0014974152 -0.5715212328  0.3493452597 
##           374           375           376           377           378 
## -0.0108281070  0.1539453990  0.1471453783  0.1363654274 -0.1761378458 
##           379           380           381           382           383 
## -1.2813513284  0.0918190206  0.5983888136  0.1428043264  0.1264316981 
##           384           385           386           387           388 
## -0.8391078744 -0.9738860292  0.5109593596  0.4516808795  0.0286400081 
##           389           390           391           392           393 
##  0.0901745677  0.1585294149  0.0841943044 -0.6683807394 -0.2030406792 
##           394           395           396           397           398 
##  0.2384969009 -0.1390712810 -0.1270707675 -0.0270231052  0.1284710760 
##           399           400           401           402           403 
##  0.0379955508 -0.0219594934 -0.1695280788 -0.1986080847  0.0334525695 
##           404           405           406           407           408 
## -0.2926204456 -0.1934232841 -0.0176278140  0.0273915475  0.0325766092 
##           409           410           411           412           413 
## -0.1181397634 -0.2282183014  0.0973438501 -0.2197736790 -0.2375855473 
##           414           415           416           417           418 
##  0.2498148389 -0.2043184430 -0.2744414909 -0.3186393395 -0.1675935113 
##           419           420           421           422           423 
## -0.0096804044 -0.5760188683  0.0793725170 -0.2573768985 -0.2480585593 
##           424           425           426           427           428 
##  0.0715408880  0.1157720377 -0.7066065351 -0.0867812840  0.0569883563 
##           429           430           431           432           433 
##  0.0947917481  0.1210998519 -0.3545813336  0.4591536598 -0.0625312590 
##           434           435           436           437           438 
## -0.0061132219 -0.2725007638  0.2364259393 -0.4822548823  0.2510001141 
##           439           440           441           442           443 
## -0.1846323061 -0.2436370927 -0.1394006311  0.3610730304  0.1408407562 
##           444           445           446           447           448 
## -0.4440536058  0.0456577577  0.3197413943 -0.1396650524 -0.3679059246 
##           449           450           451           452           453 
## -0.0301237031 -0.2638534820  0.2964331899 -0.3556811949  0.0118432188 
##           454           455           456           457           458 
##  0.5180191008 -0.0852038610  0.2190573660  0.0510070492 -0.2558389243 
##           459           460           461           462           463 
##  0.3315890975 -0.0066144724  0.1248196722  0.4019280524 -0.0511776886 
##           464           465           466           467           468 
## -0.1880112413  0.2141152837  0.2201229116  0.0967491989 -0.1180321913 
##           469           470           471           472           473 
##  0.2831865550 -0.0233968008  0.0160941665  0.1423953231 -0.1018474882 
##           474           475           476           477           478 
## -0.0092405015  0.1202430617  0.0252630344  0.2573463433 -0.3121834500 
##           479           480           481           482           483 
## -0.0028087914  0.0633561652 -0.1839499314  0.0900085241  0.0628534532 
##           484           485           486           487           488 
## -0.2220201055  0.1877685576 -0.0358576711  0.0237494221  0.2100466351 
##           489           490 
## -0.0624711829  0.3104886852

remove <- list(p =list(TAVG=c(1:14),mean_precipation=c(1:14)))
mod_ardl914_weather <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                 TAVG,data = full_cases_wastewater_weather_data_train, 
                               p=14,q=9,
                               remove = remove)
summary(mod_ardl914_weather)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.40290 -0.14197  0.01842  0.15906  1.41266 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.867954   0.335869  -2.584  0.01008 *  
## log_viral_gene.t      0.031642   0.026145   1.210  0.22682    
## log_viral_gene.1      0.012500   0.033124   0.377  0.70607    
## log_viral_gene.2     -0.023609   0.033196  -0.711  0.47733    
## log_viral_gene.3      0.025952   0.033267   0.780  0.43573    
## log_viral_gene.4     -0.002508   0.033336  -0.075  0.94007    
## log_viral_gene.5      0.034824   0.033203   1.049  0.29483    
## log_viral_gene.6      0.016574   0.033305   0.498  0.61898    
## log_viral_gene.7     -0.033090   0.033232  -0.996  0.31992    
## log_viral_gene.8     -0.012193   0.033196  -0.367  0.71357    
## log_viral_gene.9      0.009948   0.033083   0.301  0.76378    
## log_viral_gene.10    -0.002476   0.033025  -0.075  0.94028    
## log_viral_gene.11     0.051209   0.032938   1.555  0.12072    
## log_viral_gene.12    -0.060078   0.032996  -1.821  0.06931 .  
## log_viral_gene.13    -0.007915   0.033065  -0.239  0.81091    
## log_viral_gene.14     0.017932   0.026149   0.686  0.49322    
## mean_precipation.t   -0.045050   0.051399  -0.876  0.38123    
## TAVG.t                0.001150   0.001211   0.949  0.34290    
## log_mean_new_cases.1  0.476385   0.047191  10.095  < 2e-16 ***
## log_mean_new_cases.2  0.123148   0.052311   2.354  0.01899 *  
## log_mean_new_cases.3  0.089966   0.052254   1.722  0.08581 .  
## log_mean_new_cases.4  0.104632   0.052565   1.991  0.04714 *  
## log_mean_new_cases.5  0.160262   0.052376   3.060  0.00235 ** 
## log_mean_new_cases.6  0.032736   0.052929   0.618  0.53657    
## log_mean_new_cases.7  0.089809   0.052619   1.707  0.08856 .  
## log_mean_new_cases.8 -0.105577   0.052471  -2.012  0.04481 *  
## log_mean_new_cases.9 -0.071901   0.047372  -1.518  0.12977    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3364 on 449 degrees of freedom
## Multiple R-squared:  0.9135, Adjusted R-squared:  0.9085 
## F-statistic: 182.4 on 26 and 449 DF,  p-value: < 2.2e-16
f_ardl914_weather <- forecast(mod_ardl914_weather, x= t(full_cases_wastewater_weather_data_test[,c(7,4,5)]),h=14, interval = TRUE)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_ardl914_weather$forecasts[,2]) 
## [1] 0.2279778
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
    f_ardl914_weather$forecasts[,2]) 
## [1] 0.2014719
checkresiduals(mod_ardl914_weather)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
## -0.1598551047 -0.1225165616 -0.0301679289 -0.0208382419 -0.0723169834 
##            20            21            22            23            24 
##  0.1685730951  0.0070409157 -0.1692292859  0.3059135597 -0.0334645403 
##            25            26            27            28            29 
##  0.0246616435 -0.2479763972  0.3152365404 -0.2246397640 -0.2925211907 
##            30            31            32            33            34 
##  0.1511805887 -0.0572522361  0.0968126401  0.2632154121  0.1389891099 
##            35            36            37            38            39 
## -0.1126687887 -0.2812000492  0.1007599223 -0.1795412432 -0.0270208992 
##            40            41            42            43            44 
##  0.0772433990  0.0694684046 -0.2446254706 -0.0276763095 -0.0892349781 
##            45            46            47            48            49 
##  0.1347692461  0.1009308698 -0.0313555466  0.1253004386 -0.1931288974 
##            50            51            52            53            54 
##  0.1142766485 -0.1443105236 -0.0850959969 -0.3892249998  0.0114853287 
##            55            56            57            58            59 
##  0.1177683209  0.1146963395  0.1803954814 -0.0035036643 -0.2115549331 
##            60            61            62            63            64 
##  0.1523659667  0.1157369297 -0.1126361059  0.0773482648  0.0419675080 
##            65            66            67            68            69 
##  0.1963528047 -0.0041495371  0.0252418656  0.1781255163  0.0874398014 
##            70            71            72            73            74 
##  0.2761932592  0.4527801216 -0.0269581382 -0.1039303667  0.0546314086 
##            75            76            77            78            79 
## -0.3522411138  0.1072253719 -0.1134907594  0.1375515031  0.1076228497 
##            80            81            82            83            84 
##  0.0551747713 -0.2914312775  0.0200215046 -0.2016059324  0.1072610916 
##            85            86            87            88            89 
##  0.3611947058 -0.2967177734 -0.1060418715  0.2254720881 -0.2385084237 
##            90            91            92            93            94 
## -0.5006965686 -0.1273601464  0.3890071565  0.4452048659  0.1875955743 
##            95            96            97            98            99 
##  0.0798735963  0.3723132674  0.1906646577  0.1391579161  0.2348353271 
##           100           101           102           103           104 
## -0.4950425448 -0.0184920863 -0.2620771858 -0.1311337943 -0.1445183473 
##           105           106           107           108           109 
## -0.0184835067  0.0008695043  0.1410945754 -0.0955488407  0.0404119424 
##           110           111           112           113           114 
##  0.1528186645 -0.0354792912  0.2036520149  0.2533699211 -0.0102068272 
##           115           116           117           118           119 
## -0.0179557903 -0.0190902491 -0.1659055829  0.0728095109 -0.2588471366 
##           120           121           122           123           124 
##  0.3077816170 -0.2780509186  0.2029949600 -0.0425493956  0.1574347118 
##           125           126           127           128           129 
## -0.1286662381  0.0286641196  0.1526639469 -0.0077900456  0.1067838195 
##           130           131           132           133           134 
##  0.2133298961 -0.0712961699  0.2065735849 -0.0379197929  0.0470094333 
##           135           136           137           138           139 
## -0.0453414560  0.3839706948 -0.1897563313  0.0874585831 -0.2874148691 
##           140           141           142           143           144 
##  0.1806938482  0.3024595124 -0.1373243749  0.0948702093 -0.2780971851 
##           145           146           147           148           149 
## -0.2323990103 -0.2173007217 -0.0399900874  0.2498237893 -1.2596073938 
##           150           151           152           153           154 
##  0.1096847620 -1.1282296765  0.1618266313  0.4030587640  0.3794306996 
##           155           156           157           158           159 
##  0.4521777317 -0.3214800728 -0.3017426112 -0.2315349987 -0.1831764115 
##           160           161           162           163           164 
## -0.6512397564  0.1016188893 -0.1658488902  0.5320631477  0.1893867020 
##           165           166           167           168           169 
##  0.1559420518 -0.0238560448  0.2254989572 -0.1329705283 -0.2345259894 
##           170           171           172           173           174 
## -0.3688520889 -0.0016791787 -0.7335084811  0.0848262881 -0.2762469871 
##           175           176           177           178           179 
##  0.4251268976  0.1082544641 -0.0260387327 -0.3259783422 -0.1257436967 
##           180           181           182           183           184 
##  0.0727279454  0.1626047394 -0.2097714256 -1.6266709567  0.6256432427 
##           185           186           187           188           189 
##  0.1663681233  0.0911628298  0.2253879837  0.6155539970  0.1546327097 
##           190           191           192           193           194 
##  0.3140108016 -0.0548312269  0.0570171843 -0.3274630483  0.2813735459 
##           195           196           197           198           199 
## -0.1832264385 -0.0601509779  0.2036643922 -0.1216389817  0.3938849658 
##           200           201           202           203           204 
## -0.0208546847  0.3566763342  0.0879584789  0.1439236622  0.4242051741 
##           205           206           207           208           209 
##  0.1216375086 -0.0750703195 -0.1357665603  0.2579409542  0.0095267095 
##           210           211           212           213           214 
## -0.0258679035 -0.0606696151  0.1908595613 -0.0702991206 -0.0108317933 
##           215           216           217           218           219 
##  0.2019567514 -0.1472938014  0.0212717739  0.1613945777  0.0276810837 
##           220           221           222           223           224 
## -0.0231533967 -0.0461885368  0.1333367022 -0.0072548353 -0.0579945543 
##           225           226           227           228           229 
## -0.0146324119  0.1289198862 -0.0544152067  0.1197575347  0.1582838938 
##           230           231           232           233           234 
##  0.0574389441  0.0736530017 -0.1263916398  0.1917635144 -0.0563831860 
##           235           236           237           238           239 
##  0.1054594800  0.1293861842  0.1403057209  0.5255974973 -0.4081098287 
##           240           241           242           243           244 
## -0.0547901318 -0.2502529804  0.0523296813  0.1992923761 -0.0211109595 
##           245           246           247           248           249 
## -0.0526225682 -0.5923648641 -1.6034208234  0.8617166269 -0.1366033044 
##           250           251           252           253           254 
## -0.0753733026  0.1491045660  0.2866682014 -0.2503456919  0.0463365326 
##           255           256           257           258           259 
## -0.5210128224 -0.0261498079  0.0466911956 -0.0257058195  0.0630534023 
##           260           261           262           263           264 
## -0.2667616713 -0.0425015436 -0.1205978693 -0.1401521421 -0.2577285786 
##           265           266           267           268           269 
##  0.0486796698 -0.2734465328 -0.4363797992  0.3658869262 -0.1309025208 
##           270           271           272           273           274 
## -0.0603408747  0.0357190141 -0.0075485069  0.1841060560 -0.4372974589 
##           275           276           277           278           279 
## -0.1452598839 -0.2709641177  0.0066672901 -0.1035533810  0.8509232636 
##           280           281           282           283           284 
## -0.4899937118  0.4447536103 -0.2536189764  0.0103402816  0.4222617360 
##           285           286           287           288           289 
##  0.0182503708 -0.4878893905 -0.2486933619 -0.7841699623  0.2914797862 
##           290           291           292           293           294 
## -0.1161855048  0.1079270656 -2.4028952730 -0.4205152574  1.4126643769 
##           295           296           297           298           299 
## -0.2010308451  0.3252210579  0.5911258621  0.4199844056  0.2029666495 
##           300           301           302           303           304 
##  0.1026688446 -0.3113853555 -0.5640342023 -0.2255618647 -0.2302023467 
##           305           306           307           308           309 
## -0.0848028062 -0.3072288604  0.1421825110  0.0295385321  0.4653148025 
##           310           311           312           313           314 
## -0.2747705195 -0.9408806242  0.4791519963 -0.2958831994  0.4518559305 
##           315           316           317           318           319 
## -0.0933355497  0.2796988693  0.2059272145 -0.0487177701  0.1105670415 
##           320           321           322           323           324 
##  0.0185848795  0.1100495594  0.4638978025  0.0464922528  0.0614714762 
##           325           326           327           328           329 
## -0.0398100947 -0.0454480745 -0.6917372255  0.2288681765  0.3172829818 
##           330           331           332           333           334 
## -0.0591268650  0.4968998786  0.2243777869  0.5257070882  0.0580607165 
##           335           336           337           338           339 
##  0.0884957742  0.0187205155  0.2638504302 -0.1418219020  0.0759945435 
##           340           341           342           343           344 
## -0.2198768254 -0.3543629620  0.1378017026  0.0057710941  0.2672497147 
##           345           346           347           348           349 
##  0.0418643909 -0.1568855261 -0.0883063410  0.0643899230  0.0949904953 
##           350           351           352           353           354 
##  0.4701213379  0.2950359043  0.1446385387  0.3203232057  0.3339262169 
##           355           356           357           358           359 
##  0.5455419438 -0.2966028204 -0.5296154232  0.9101102717  0.9039664027 
##           360           361           362           363           364 
##  0.6830922666  0.5106395635  0.4811403427 -0.1424008571  0.0514878225 
##           365           366           367           368           369 
## -0.0136667424  0.1887897764  0.4792338088  0.3156212166  0.2533467269 
##           370           371           372           373           374 
##  0.1673046628  0.0330032092 -0.5747318501  0.3291672224 -0.0174419297 
##           375           376           377           378           379 
##  0.1562413692  0.1701293672  0.1575541081 -0.1540578396 -1.2564935345 
##           380           381           382           383           384 
##  0.0962594584  0.5809421085  0.1709079095  0.1414251990 -0.8290379833 
##           385           386           387           388           389 
## -0.9572761094  0.5360977090  0.5092223530 -0.0466106074  0.1098035736 
##           390           391           392           393           394 
##  0.1769070473  0.1132475104 -0.6218903338 -0.2360907553  0.1207579338 
##           395           396           397           398           399 
## -0.1643837006 -0.0825364288  0.0167049010  0.1356470150  0.0507710700 
##           400           401           402           403           404 
## -0.0109159064 -0.1758022956 -0.2472423085  0.0739724459 -0.2870882873 
##           405           406           407           408           409 
## -0.1967654875 -0.0186030947  0.0141095675  0.0593044169 -0.1041403063 
##           410           411           412           413           414 
## -0.2201607022  0.0965375047 -0.1979194174 -0.2334797314  0.2433807593 
##           415           416           417           418           419 
## -0.2088143299 -0.2639571698 -0.2999839483 -0.1712260252 -0.0211648734 
##           420           421           422           423           424 
## -0.5619845374  0.0673473514 -0.2613870199 -0.2327995888  0.0962848637 
##           425           426           427           428           429 
##  0.1270484552 -0.7137764687 -0.0931071008  0.0626497513  0.0742202975 
##           430           431           432           433           434 
##  0.1454931697 -0.3461678461  0.4490582422 -0.0496427066  0.0160140915 
##           435           436           437           438           439 
## -0.2986167052  0.2055632679 -0.4791739946  0.2187411454 -0.1364953791 
##           440           441           442           443           444 
## -0.2721438772 -0.1104588575  0.4045871484  0.1435415498 -0.4474806112 
##           445           446           447           448           449 
##  0.0405657595  0.3003622089 -0.1384350711 -0.3655598330 -0.0214074404 
##           450           451           452           453           454 
## -0.2978741639  0.2919823929 -0.3359265953 -0.0201083441  0.5072172010 
##           455           456           457           458           459 
## -0.0579247087  0.2381631998  0.0244983684 -0.2372769165  0.2924483937 
##           460           461           462           463           464 
## -0.0022931343  0.0958687563  0.3655569702 -0.0456459814 -0.2235273489 
##           465           466           467           468           469 
##  0.2225173991  0.2103193547  0.0623996137 -0.1172026721  0.2644340519 
##           470           471           472           473           474 
## -0.0437126069  0.0280937499  0.1418535921 -0.1314913335 -0.0261384617 
##           475           476           477           478           479 
##  0.1250681381  0.0261809836  0.2274880449 -0.2989033067  0.0026326635 
##           480           481           482           483           484 
##  0.0433829328 -0.1726916006  0.0832219875  0.0580524936 -0.2144775035 
##           485           486           487           488           489 
##  0.1802469574 -0.0018098246 -0.0056430255  0.2060659052 -0.0612018247 
##           490 
##  0.2915620203

exp(f_ardl914_weather$forecasts[1,2])
## [1] 5.367364
exp(f_ardl914_weather$forecasts[1,1])
## [1] 2.946514
exp(f_ardl914_weather$forecasts[1,3])
## [1] 9.605158
exp(f_ardl914_weather$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_test[1,6])
## [1] -3.35018
exp(f_ardl914_weather$forecasts[7,2])
## [1] 5.95613
exp(f_ardl914_weather$forecasts[7,1])
## [1] 2.650773
exp(f_ardl914_weather$forecasts[7,3])
## [1] 14.93228
exp(f_ardl914_weather$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_test[7,6])
## [1] -1.042123
exp(f_ardl914_weather$forecasts[14,2])
## [1] 7.352754
exp(f_ardl914_weather$forecasts[14,1])
## [1] 2.502025
exp(f_ardl914_weather$forecasts[14,3])
## [1] 23.27684
exp(f_ardl914_weather$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_test[14,6])
## [1] 1.646034
#Mecklenburg

full_cases_wastewater_weather_data_meck <- full_cases_wastewater_weather_data_meck[-c(505,506,507),]

full_cases_wastewater_weather_data_meck <- full_cases_wastewater_weather_data_meck %>% 
  mutate(log_mean_new_cases = log(mean_new_cases),
         log_viral_gene = log(full_viral_gene_copies_per_person))

full_cases_wastewater_weather_data_meck <- full_cases_wastewater_weather_data_meck %>% 
  mutate(log_mean_new_cases = seasadj(decompose(ts(log_mean_new_cases, frequency=7))),
         log_viral_gene = seasadj(decompose(ts(log_viral_gene, frequency=7))))


full_cases_wastewater_weather_data_meck_train <- 
  full_cases_wastewater_weather_data_meck[-c(491:504),]
full_cases_wastewater_weather_data_meck_test <- 
  full_cases_wastewater_weather_data_meck[c(491:504),]

lowest_rmse_meck <- Inf
best_mod_meck <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                   data = full_cases_wastewater_weather_data_meck_train, p=p,q=q)
    f <- forecast(mod, x= t(full_cases_wastewater_weather_data_meck_test[,8]),h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
                         f$forecasts) #interchanged between RMSE and MAE 
    if (forecast_acc<lowest_rmse_meck){
      lowest_rmse_meck<- forecast_acc
      best_mod_meck <-mod 
    }
  }
}

lowest_rmse_meck #0.11
## [1] 0.1107693
summary(best_mod_meck) #ARDL(5,1)
## 
## Time series regression with "ts" data:
## Start = 6, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7591 -0.1596  0.0036  0.1817  1.1136 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.83391    0.31714  -2.629 0.008829 ** 
## log_viral_gene.t     -0.04353    0.03451  -1.261 0.207770    
## log_viral_gene.1      0.09913    0.03463   2.863 0.004383 ** 
## log_mean_new_cases.1  0.52109    0.04530  11.502  < 2e-16 ***
## log_mean_new_cases.2  0.06677    0.05100   1.309 0.191121    
## log_mean_new_cases.3  0.17238    0.05038   3.421 0.000677 ***
## log_mean_new_cases.4  0.06100    0.05080   1.201 0.230452    
## log_mean_new_cases.5  0.10279    0.04502   2.283 0.022851 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3132 on 477 degrees of freedom
## Multiple R-squared:  0.9209, Adjusted R-squared:  0.9198 
## F-statistic: 793.6 on 7 and 477 DF,  p-value: < 2.2e-16
mod_ardl51_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
               data = full_cases_wastewater_weather_data_meck_train, p=1,q=5)
summary(mod_ardl51_meck) #wastewater is significant at lagged time t-1
## 
## Time series regression with "ts" data:
## Start = 6, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7591 -0.1596  0.0036  0.1817  1.1136 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.83391    0.31714  -2.629 0.008829 ** 
## log_viral_gene.t     -0.04353    0.03451  -1.261 0.207770    
## log_viral_gene.1      0.09913    0.03463   2.863 0.004383 ** 
## log_mean_new_cases.1  0.52109    0.04530  11.502  < 2e-16 ***
## log_mean_new_cases.2  0.06677    0.05100   1.309 0.191121    
## log_mean_new_cases.3  0.17238    0.05038   3.421 0.000677 ***
## log_mean_new_cases.4  0.06100    0.05080   1.201 0.230452    
## log_mean_new_cases.5  0.10279    0.04502   2.283 0.022851 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3132 on 477 degrees of freedom
## Multiple R-squared:  0.9209, Adjusted R-squared:  0.9198 
## F-statistic: 793.6 on 7 and 477 DF,  p-value: < 2.2e-16
f_ardl51_meck <- forecast(mod_ardl51_meck , x= t(full_cases_wastewater_weather_data_meck_test[,8]),h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
                     f_ardl51_meck$forecasts)
## [1] 0.1107693
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl51_meck$forecasts)
## [1] 0.09374707
checkresiduals(mod_ardl51_meck)
## Time Series:
## Start = 6 
## End = 490 
## Frequency = 1 
##            6            7            8            9           10           11 
## -0.144992803 -0.154528950 -0.133926698  0.182971537 -0.054586470 -0.025672398 
##           12           13           14           15           16           17 
##  0.003600857  0.089782491 -0.004708061  0.025199802 -0.003101377 -0.116164706 
##           18           19           20           21           22           23 
## -0.168196341 -0.024364651  0.222029565  0.047886939 -0.050931023 -0.138940545 
##           24           25           26           27           28           29 
##  0.214465000  0.025827523 -0.225926344 -0.070952987  0.202837773 -0.016761819 
##           30           31           32           33           34           35 
## -0.026726462  0.014734252 -0.224967432 -0.059531178 -0.302987185  0.330717347 
##           36           37           38           39           40           41 
##  0.033608659 -0.135211554 -0.124342468 -0.240078271  0.055652050  0.118961867 
##           42           43           44           45           46           47 
## -0.088270625 -0.166181236  0.094532619 -0.106232081 -0.250429346  0.103190486 
##           48           49           50           51           52           53 
## -0.256810581  0.303153163 -0.235929863 -0.415643589  0.128535816  0.006584904 
##           54           55           56           57           58           59 
## -0.005906975 -0.078568003 -0.140365117  0.327786636 -0.343437022  0.156674417 
##           60           61           62           63           64           65 
## -1.036322792 -0.274784371  0.195454625 -0.565710697  0.590719314  0.363041449 
##           66           67           68           69           70           71 
##  0.099115304 -0.067479514  0.278631327  0.257900498  0.414386778 -0.043795543 
##           72           73           74           75           76           77 
## -0.007374073 -0.257995332  0.404066428 -0.156858879  0.470228526  0.312996989 
##           78           79           80           81           82           83 
##  0.295946934 -0.060940902 -0.010522861 -0.119146048 -0.254959730  0.001253895 
##           84           85           86           87           88           89 
##  0.009725621  0.336664566 -0.246171855  0.359536395  0.003037467  0.201517562 
##           90           91           92           93           94           95 
## -0.234612105 -0.032931510  0.027308133 -0.350542165  0.027188701 -0.179031042 
##           96           97           98           99          100          101 
##  0.151804344  0.064175256 -0.022141232 -0.039675842 -0.094652727  0.193928855 
##          102          103          104          105          106          107 
## -0.079727028 -0.353565133  0.366210572 -0.476732118  0.592983031 -0.066852213 
##          108          109          110          111          112          113 
## -0.135521183  0.095786686 -0.297134455  0.070047707  0.348012742 -0.131628833 
##          114          115          116          117          118          119 
## -0.011948338 -0.144414468 -0.440057988  0.331071631 -0.299400679 -0.070169163 
##          120          121          122          123          124          125 
## -0.130101619 -0.007333190 -0.187350486 -0.020263682 -0.360826411  0.017733707 
##          126          127          128          129          130          131 
##  0.289677054 -0.272543064 -0.666636399 -0.177430741 -0.898830212  0.110702455 
##          132          133          134          135          136          137 
## -0.143884596  0.778765330 -0.011133061  0.047667249 -0.372625096 -0.263415463 
##          138          139          140          141          142          143 
##  0.126560172 -0.427779448 -0.162454304  0.418718268 -0.300359969 -0.579639640 
##          144          145          146          147          148          149 
##  0.438529036  0.430950455  0.078395787  0.089242473 -0.207584876 -0.321396510 
##          150          151          152          153          154          155 
##  0.935631706  0.024385520 -0.654667871 -0.320742908  0.624255405 -0.438734676 
##          156          157          158          159          160          161 
## -0.188115833 -0.190563186  0.389737781  0.147281439 -0.370012084  0.344793461 
##          162          163          164          165          166          167 
##  0.386640243  0.278368538  0.236444776 -0.289484029 -0.239347323 -0.147775084 
##          168          169          170          171          172          173 
##  0.421903669 -0.472504641 -0.592796447 -0.398360812  0.205100174  0.015503158 
##          174          175          176          177          178          179 
## -0.748619246  0.366642728  0.207370920 -0.329533286  0.039462191 -0.348921816 
##          180          181          182          183          184          185 
##  0.902647851  0.239740997  0.136067113 -0.403975534  0.076238215 -0.257722454 
##          186          187          188          189          190          191 
##  0.196253637  0.510291413  0.513786053  0.521666601  0.225189039 -0.206571379 
##          192          193          194          195          196          197 
##  0.218983696 -0.120526134  0.056688808  0.151611857  0.016239988  0.705324127 
##          198          199          200          201          202          203 
## -0.030912129  0.173137513  0.199105669  0.160386211  0.304621289  0.242804904 
##          204          205          206          207          208          209 
##  0.216044301  0.091175723 -0.099425313  0.232585680  0.110244477  0.016836381 
##          210          211          212          213          214          215 
##  0.206028007  0.202335411  0.057424217 -0.172499638 -0.046289502  0.177043374 
##          216          217          218          219          220          221 
## -0.022717898 -0.137142897  0.182413491 -0.035504820  0.040322229  0.192119224 
##          222          223          224          225          226          227 
##  0.051986352  0.099150905 -0.256522770 -0.017478788  0.005780314  0.126185096 
##          228          229          230          231          232          233 
##  0.085020801  0.125825516  0.157224875  0.037575411  0.181811293  0.160002425 
##          234          235          236          237          238          239 
## -0.144176437 -0.132360134  0.005312516  0.155005045 -0.050772582 -0.061820079 
##          240          241          242          243          244          245 
##  0.130881559 -0.118542212 -0.265687059  0.100253328 -0.026092595 -0.105632762 
##          246          247          248          249          250          251 
##  0.006946720 -0.855422488  0.489696495  0.182152765  0.183784260  0.072996289 
##          252          253          254          255          256          257 
##  0.092604453 -0.207437140  0.113043773 -0.118922586 -0.309382751 -0.005559694 
##          258          259          260          261          262          263 
##  0.062175309 -0.151915285 -0.209386914  0.041819824 -0.146114434 -0.231540655 
##          264          265          266          267          268          269 
##  0.064320055 -0.131629585 -0.048747819 -0.288049131  0.103739890 -0.252829716 
##          270          271          272          273          274          275 
##  0.391770669 -0.065970454  0.131360564 -0.179354336 -0.306677122  0.045238891 
##          276          277          278          279          280          281 
## -0.059912722 -0.397106445 -0.094034469  0.079267901 -0.303629835  0.024189968 
##          282          283          284          285          286          287 
## -0.022693476 -0.143050897  0.073425525 -0.136597452  0.168396949  0.053725800 
##          288          289          290          291          292          293 
## -0.147249882 -0.147967309 -0.364078434 -0.125688530 -1.196700945 -0.282673542 
##          294          295          296          297          298          299 
##  0.670515901  0.533289970 -0.028610032  0.101910953  0.234739102 -0.406847444 
##          300          301          302          303          304          305 
##  0.189635572  0.264275301 -0.166644696 -0.002768102  0.008631935 -0.570484138 
##          306          307          308          309          310          311 
##  0.249815590 -0.073528528  0.200842317  0.043590647  0.075978870 -0.262761353 
##          312          313          314          315          316          317 
##  0.025882028 -0.030714663  0.122142094  0.096363475  0.284255533 -0.135206941 
##          318          319          320          321          322          323 
##  0.197810465  0.021309613  0.049463195 -0.073689175  0.151796019 -0.046817494 
##          324          325          326          327          328          329 
## -0.138715949  0.205704005 -0.024223650 -0.513306660  0.298996982  0.805954156 
##          330          331          332          333          334          335 
##  0.021648674  0.308777596 -0.065833168  0.212915000  0.041421806  0.473468117 
##          336          337          338          339          340          341 
## -0.110225950 -0.172449698  0.099445579 -0.134915004 -0.024795992 -0.047601090 
##          342          343          344          345          346          347 
##  0.124022800  0.068010871 -0.004663416  0.320403573  0.027263164  0.275705782 
##          348          349          350          351          352          353 
##  0.245434695  0.361830532  0.180135636  0.267686172  0.264712533  0.381481079 
##          354          355          356          357          358          359 
##  0.429917116  0.399063104  0.013212848 -0.665136847  0.811976751  0.653863118 
##          360          361          362          363          364          365 
##  0.627484178  0.478597552  0.479212778 -0.112929716 -0.244679814  0.353230629 
##          366          367          368          369          370          371 
##  0.390134188  0.312537387  0.299665291  0.174226975 -0.072554462 -0.294971350 
##          372          373          374          375          376          377 
## -0.144349298  0.276618918  0.013716003  0.104845559  0.048590437  0.073137496 
##          378          379          380          381          382          383 
## -0.288019052 -1.759116418  0.238733695  0.548555331  0.425413105  0.298553523 
##          384          385          386          387          388          389 
## -0.140345543 -0.542320158 -0.031626245  0.088830281 -0.090736332 -0.068731086 
##          390          391          392          393          394          395 
## -0.111162495 -0.196964594 -0.319511537 -0.054113044  0.149243114 -0.278231170 
##          396          397          398          399          400          401 
##  0.107463973 -0.235278606  0.042654836 -0.314788705 -0.159564779  0.165631549 
##          402          403          404          405          406          407 
## -0.124195848  0.021199812 -0.074856428  0.003123220 -0.194514790 -0.599461655 
##          408          409          410          411          412          413 
##  0.181716654 -0.365029118 -0.201435520 -0.169902263 -0.025851496 -0.076552159 
##          414          415          416          417          418          419 
##  0.088030763 -0.099582205 -0.043695050 -0.269273042 -0.275890575  0.453215587 
##          420          421          422          423          424          425 
## -0.648014612 -1.039674707  0.313312115 -0.458413319 -0.243757085  0.043610984 
##          426          427          428          429          430          431 
## -0.409536010 -0.647201886  0.725509043 -0.526357990 -0.257236978 -0.320068168 
##          432          433          434          435          436          437 
## -0.747243395  0.126461863  0.151211756 -0.195650731 -0.435941417 -0.033441117 
##          438          439          440          441          442          443 
##  0.887763414 -0.354115210  0.066950153 -0.297217749  0.114176726  0.248600861 
##          444          445          446          447          448          449 
## -0.237119057  0.171923712  0.152815884 -0.274514800 -0.442306864  0.528678067 
##          450          451          452          453          454          455 
## -0.700643002  1.113551041  0.124461468 -0.002252850 -0.439067637  0.045817052 
##          456          457          458          459          460          461 
## -0.055935982  0.208110401 -0.026114953  0.292336475  0.195834414 -0.154004131 
##          462          463          464          465          466          467 
## -0.561091450  0.475967668  0.501946529  0.212726944  0.339048541  0.361194746 
##          468          469          470          471          472          473 
## -0.414063098 -0.038215609 -0.612293382  0.434307608  0.109741430  0.048668491 
##          474          475          476          477          478          479 
##  0.153398245 -0.196706260 -0.356587010  0.067284042  0.504891508 -0.059144656 
##          480          481          482          483          484          485 
##  0.228707726 -0.041757265 -0.061190527  0.050164468  0.042556326  0.204656551 
##          486          487          488          489          490 
## -0.025643610  0.162770989  0.068308710  0.002701517  0.155538875

mod_ardl84_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                      data = full_cases_wastewater_weather_data_meck_train,
                      p=4,q=8)
summary(mod_ardl84_meck)
## 
## Time series regression with "ts" data:
## Start = 9, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.78380 -0.15859 -0.00029  0.17472  1.13456 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.64873    0.36787  -1.763 0.078471 .  
## log_viral_gene.t     -0.05323    0.03506  -1.518 0.129652    
## log_viral_gene.1      0.10831    0.04532   2.390 0.017239 *  
## log_viral_gene.2     -0.03948    0.04551  -0.868 0.386111    
## log_viral_gene.3      0.11034    0.04559   2.420 0.015884 *  
## log_viral_gene.4     -0.08245    0.03506  -2.351 0.019112 *  
## log_mean_new_cases.1  0.52497    0.04610  11.387  < 2e-16 ***
## log_mean_new_cases.2  0.06068    0.05195   1.168 0.243404    
## log_mean_new_cases.3  0.19410    0.05192   3.738 0.000208 ***
## log_mean_new_cases.4  0.06080    0.05200   1.169 0.242904    
## log_mean_new_cases.5  0.09745    0.05179   1.882 0.060515 .  
## log_mean_new_cases.6  0.06023    0.05126   1.175 0.240619    
## log_mean_new_cases.7 -0.01669    0.05107  -0.327 0.743978    
## log_mean_new_cases.8 -0.04549    0.04586  -0.992 0.321746    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3129 on 468 degrees of freedom
## Multiple R-squared:  0.9221, Adjusted R-squared:  0.9199 
## F-statistic: 426.1 on 13 and 468 DF,  p-value: < 2.2e-16
f_ardl84_meck <- forecast(mod_ardl84_meck , 
                          x= t(full_cases_wastewater_weather_data_meck_test[,8]),
                          h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl84_meck$forecasts)
## [1] 0.1121279
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl84_meck$forecasts)
## [1] 0.0923158
checkresiduals(mod_ardl84_meck)
## Time Series:
## Start = 9 
## End = 490 
## Frequency = 1 
##             9            10            11            12            13 
##  2.114865e-01 -6.412026e-02  5.053977e-02 -4.626028e-02  9.601234e-02 
##            14            15            16            17            18 
##  4.090677e-03  1.506730e-02  1.295953e-03 -1.121956e-01 -1.720069e-01 
##            19            20            21            22            23 
## -5.737256e-02  2.566891e-01  5.100448e-02 -4.963672e-02 -1.324656e-01 
##            24            25            26            27            28 
##  2.314733e-01 -2.187509e-05 -2.651579e-01 -4.760945e-02  2.094540e-01 
##            29            30            31            32            33 
## -1.507552e-02 -3.481667e-02 -2.679562e-02 -1.458731e-01 -2.444237e-02 
##            34            35            36            37            38 
## -4.094079e-01  3.307782e-01  4.719045e-02 -1.102542e-01 -1.005338e-01 
##            39            40            41            42            43 
## -2.676435e-01  5.671593e-02  1.286972e-01 -9.334871e-02 -1.475369e-01 
##            44            45            46            47            48 
##  1.066811e-01 -9.330449e-02 -2.613879e-01  1.211922e-01 -2.684318e-01 
##            49            50            51            52            53 
##  3.285657e-01 -2.391886e-01 -4.018863e-01  1.693160e-01 -5.568607e-02 
##            54            55            56            57            58 
## -2.313547e-02 -1.694647e-02 -1.356441e-01  3.580371e-01 -3.509830e-01 
##            59            60            61            62            63 
##  1.273845e-01 -9.987574e-01 -2.594677e-01  1.720282e-01 -5.704496e-01 
##            64            65            66            67            68 
##  5.939196e-01  3.676968e-01  6.256490e-02  1.217482e-01  2.168790e-01 
##            69            70            71            72            73 
##  2.556702e-01  3.590379e-01 -9.562880e-02 -1.700112e-02 -1.586703e-01 
##            74            75            76            77            78 
##  1.583908e-01 -1.798964e-01  4.527362e-01  3.285885e-01  3.098914e-01 
##            79            80            81            82            83 
## -5.546003e-02 -3.923615e-02 -1.424467e-01 -2.703238e-01 -4.731195e-02 
##            84            85            86            87            88 
##  3.022135e-03  3.533337e-01 -2.273971e-01  3.777404e-01 -1.873153e-02 
##            89            90            91            92            93 
##  1.863850e-01 -2.208664e-01 -5.344728e-02  2.058851e-02 -3.535054e-01 
##            94            95            96            97            98 
## -1.115081e-02 -1.180501e-01  1.903608e-01 -4.033137e-03 -2.619214e-02 
##            99           100           101           102           103 
## -2.682733e-02 -7.661468e-02  1.738754e-01 -4.873295e-02 -3.699302e-01 
##           104           105           106           107           108 
##  3.505812e-01 -4.764005e-01  6.032767e-01 -8.084726e-02 -1.440491e-01 
##           109           110           111           112           113 
##  1.318371e-01 -3.443232e-01  9.853675e-02  3.201137e-01 -1.583409e-01 
##           114           115           116           117           118 
##  1.797709e-02 -1.638852e-01 -4.143736e-01  3.578324e-01 -4.115361e-01 
##           119           120           121           122           123 
## -4.911202e-02 -1.236707e-01  1.852453e-02 -1.762155e-01  1.838990e-02 
##           124           125           126           127           128 
## -3.910660e-01  2.462672e-02  2.955379e-01 -2.829274e-01 -6.436189e-01 
##           129           130           131           132           133 
## -2.607332e-01 -7.460666e-01  7.716380e-02 -1.372747e-01  7.826791e-01 
##           134           135           136           137           138 
##  3.990627e-02  6.306165e-02 -3.154304e-01 -3.635332e-01  1.025547e-01 
##           139           140           141           142           143 
## -4.772706e-01 -1.595434e-01  4.191378e-01 -2.708711e-01 -5.999573e-01 
##           144           145           146           147           148 
##  5.191558e-01  4.535474e-01  5.996607e-03 -9.373734e-04 -1.311674e-01 
##           149           150           151           152           153 
## -2.715960e-01  8.935407e-01 -3.683710e-02 -6.806805e-01 -2.365659e-01 
##           154           155           156           157           158 
##  3.931969e-01 -2.904746e-01 -2.323936e-01 -1.949843e-01  3.921652e-01 
##           159           160           161           162           163 
##  1.751432e-01 -4.108737e-01  3.422875e-01  4.240908e-01  2.947040e-01 
##           164           165           166           167           168 
##  1.910702e-01 -3.073656e-01 -2.204561e-01 -1.622853e-01  4.157310e-01 
##           169           170           171           172           173 
## -5.628103e-01 -5.950774e-01 -4.442669e-01  4.268981e-01 -8.790413e-02 
##           174           175           176           177           178 
## -7.760961e-01  3.822863e-01  2.410385e-01 -2.177960e-01  7.856436e-03 
##           179           180           181           182           183 
## -3.516450e-01  8.835614e-01  2.373113e-01  1.152154e-01 -4.077946e-01 
##           184           185           186           187           188 
##  1.237408e-01 -2.551678e-01  1.452853e-01  4.536800e-01  5.372786e-01 
##           189           190           191           192           193 
##  5.576472e-01  2.434842e-01 -2.687440e-01  2.015559e-01 -1.731056e-01 
##           194           195           196           197           198 
##  1.859901e-02  1.327086e-01  8.714665e-03  7.597549e-01 -9.719030e-02 
##           199           200           201           202           203 
##  1.777937e-01  1.753121e-01  1.634210e-01  3.103756e-01  1.962344e-01 
##           204           205           206           207           208 
##  2.355720e-01  2.579447e-02 -1.101483e-01  1.957471e-01  1.295667e-01 
##           209           210           211           212           213 
##  9.326177e-03  1.921895e-01  2.250625e-01  1.095405e-02 -1.748162e-01 
##           214           215           216           217           218 
## -4.860877e-02  1.505003e-01 -2.246712e-02 -1.416807e-01  1.745139e-01 
##           219           220           221           222           223 
##  3.175889e-03  5.633965e-02  1.416178e-01  9.650440e-02  1.124607e-01 
##           224           225           226           227           228 
## -2.674600e-01 -2.975700e-03 -4.083925e-02  1.202068e-01  6.441830e-02 
##           229           230           231           232           233 
##  1.599225e-01  1.702530e-01  4.276334e-02  1.619046e-01  1.660944e-01 
##           234           235           236           237           238 
## -1.505866e-01 -1.262807e-01 -5.238170e-02  1.587515e-01 -4.858967e-02 
##           239           240           241           242           243 
## -7.101033e-02  1.533114e-01 -1.020966e-01 -2.647150e-01  7.118711e-02 
##           244           245           246           247           248 
## -1.746871e-02 -8.212377e-02  2.090634e-02 -8.762507e-01  5.272292e-01 
##           249           250           251           252           253 
##  1.771021e-01  1.944207e-01  8.770354e-02  9.505982e-02 -1.673955e-01 
##           254           255           256           257           258 
##  1.288901e-01 -1.698953e-01 -2.957430e-01 -1.190650e-02  6.940479e-02 
##           259           260           261           262           263 
## -1.256510e-01 -1.967484e-01  2.719455e-02 -1.120842e-01 -2.413502e-01 
##           264           265           266           267           268 
##  9.046668e-02 -1.237778e-01 -2.077808e-02 -3.022515e-01  1.358665e-01 
##           269           270           271           272           273 
## -2.315067e-01  3.846026e-01 -6.256331e-02  1.496557e-01 -1.655589e-01 
##           274           275           276           277           278 
## -2.937391e-01  2.144330e-02 -6.932071e-02 -4.072256e-01 -6.913784e-02 
##           279           280           281           282           283 
##  8.298331e-02 -2.679270e-01 -2.504043e-03  2.889684e-02 -1.133434e-01 
##           284           285           286           287           288 
##  6.947883e-02 -1.471743e-01  1.864245e-01  5.975837e-02 -1.414465e-01 
##           289           290           291           292           293 
## -1.663160e-01 -3.660933e-01 -1.321922e-01 -1.188109e+00 -2.791208e-01 
##           294           295           296           297           298 
##  6.854606e-01  5.741734e-01 -5.503088e-04  1.085782e-01  2.745158e-01 
##           299           300           301           302           303 
## -3.659694e-01  1.154215e-01  2.014748e-01 -1.687743e-01  7.009330e-02 
##           304           305           306           307           308 
## -1.790854e-02 -5.183298e-01  1.614229e-01 -1.051765e-01  2.348463e-01 
##           309           310           311           312           313 
##  3.757950e-02  1.259562e-01 -2.366409e-01  4.863608e-04 -4.600879e-02 
##           314           315           316           317           318 
##  1.243534e-01  9.245526e-02  2.990036e-01 -1.568415e-01  2.043733e-01 
##           319           320           321           322           323 
## -1.871971e-02  7.720671e-02 -8.869464e-02  1.295716e-01 -4.715174e-02 
##           324           325           326           327           328 
## -1.478601e-01  1.942765e-01 -4.834625e-02 -4.917373e-01  2.841493e-01 
##           329           330           331           332           333 
##  7.934580e-01  3.316665e-02  3.080243e-01 -9.582168e-02  2.671148e-01 
##           334           335           336           337           338 
## -5.068043e-02  4.109623e-01 -1.202856e-01 -1.374851e-01  5.283613e-02 
##           339           340           341           342           343 
## -1.393160e-01 -1.524689e-02 -1.045450e-01  1.297566e-01  1.024932e-01 
##           344           345           346           347           348 
## -2.708716e-02  3.712759e-01  2.733485e-02  2.673658e-01  2.252898e-01 
##           349           350           351           352           353 
##  3.598315e-01  1.802376e-01  2.767110e-01  1.843797e-01  3.751562e-01 
##           354           355           356           357           358 
##  3.924934e-01  3.948218e-01 -6.616482e-03 -6.871730e-01  7.818524e-01 
##           359           360           361           362           363 
##  6.263925e-01  6.146656e-01  4.405028e-01  4.513664e-01 -8.045035e-02 
##           364           365           366           367           368 
## -3.001169e-01  2.543205e-01  3.740193e-01  2.972932e-01  3.435671e-01 
##           369           370           371           372           373 
##  3.372868e-02 -4.622732e-02 -2.679329e-01 -2.837823e-01  3.340631e-01 
##           374           375           376           377           378 
##  1.160161e-02  8.812509e-02  9.378014e-02  9.290555e-02 -2.734476e-01 
##           379           380           381           382           383 
## -1.783797e+00  2.354539e-01  5.548661e-01  4.596101e-01  3.055552e-01 
##           384           385           386           387           388 
## -1.314264e-01 -4.278442e-01 -3.798545e-02  2.477004e-02 -1.090278e-01 
##           389           390           391           392           393 
## -7.210254e-02 -8.451534e-02 -1.464771e-01 -3.089769e-01 -8.065425e-02 
##           394           395           396           397           398 
##  1.664003e-01 -2.551375e-01  7.743199e-02 -1.560038e-01  5.958107e-02 
##           399           400           401           402           403 
## -3.158753e-01 -1.857114e-01  1.754412e-01 -1.150685e-01 -2.463172e-02 
##           404           405           406           407           408 
##  7.116496e-03  9.622965e-03 -1.892348e-01 -5.970879e-01  1.088046e-01 
##           409           410           411           412           413 
## -3.529074e-01 -2.162829e-01 -1.229156e-01 -1.825564e-02 -3.704000e-02 
##           414           415           416           417           418 
##  8.238300e-02 -1.213248e-01 -1.790385e-02 -3.361080e-01 -1.721888e-01 
##           419           420           421           422           423 
##  4.501606e-01 -6.666377e-01 -9.880836e-01  1.987011e-01 -4.305947e-01 
##           424           425           426           427           428 
## -2.136326e-01  4.687347e-02 -3.891859e-01 -5.504740e-01  6.857461e-01 
##           429           430           431           432           433 
## -4.906559e-01 -2.077706e-01 -3.272321e-01 -7.705121e-01  1.902980e-01 
##           434           435           436           437           438 
##  1.156489e-01 -1.802610e-01 -3.750868e-01 -2.266351e-02  8.801317e-01 
##           439           440           441           442           443 
## -2.729067e-01  4.631069e-02 -3.079193e-01  1.982456e-01  1.451072e-01 
##           444           445           446           447           448 
## -2.874215e-01  1.725205e-01  1.944864e-01 -2.601933e-01 -4.456659e-01 
##           449           450           451           452           453 
##  4.819925e-01 -6.545991e-01  1.134555e+00  1.016110e-01 -1.077424e-02 
##           454           455           456           457           458 
## -4.111305e-01  6.553535e-03 -2.531074e-02  1.692089e-01 -3.413132e-02 
##           459           460           461           462           463 
##  2.327285e-01  2.771341e-01 -1.470341e-01 -5.716407e-01  4.615979e-01 
##           464           465           466           467           468 
##  4.827980e-01  2.222693e-01  3.113167e-01  3.816857e-01 -3.726330e-01 
##           469           470           471           472           473 
## -1.242595e-03 -7.860091e-01  4.189513e-01  1.261510e-01  4.027773e-02 
##           474           475           476           477           478 
##  2.257368e-01 -1.861743e-01 -3.226268e-01  3.436967e-02  5.145833e-01 
##           479           480           481           482           483 
## -3.705241e-02  2.356560e-01 -5.891424e-02 -2.200512e-02  5.080037e-02 
##           484           485           486           487           488 
##  1.972367e-02  1.825073e-01 -1.186804e-02  1.747943e-01  6.280529e-02 
##           489           490 
##  4.042257e-03  1.620899e-01

mod_ardl113_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                           data = full_cases_wastewater_weather_data_meck_train, 
                           p=13,q=1)
f_ardl113_meck <- forecast(mod_ardl113_meck , 
                           x= t(full_cases_wastewater_weather_data_meck_test[,8]),
                           h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl113_meck$forecasts)
## [1] 0.2036014
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl113_meck$forecasts)
## [1] 0.1740159
checkresiduals(mod_ardl113_meck)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
## -0.0674636102  0.0860300127  0.0262707244 -0.0927458274 -0.0712622528 
##            19            20            21            22            23 
##  0.0888639880  0.2807548902 -0.0003207917 -0.0417452450 -0.1006078488 
##            24            25            26            27            28 
##  0.3189435585  0.0092949894 -0.2308598513  0.0911981481  0.3131976662 
##            29            30            31            32            33 
## -0.0211669333  0.0342510646  0.0812954738 -0.1124101803  0.0309032328 
##            34            35            36            37            38 
## -0.2758549460  0.5008353835 -0.0333697667 -0.1656519015 -0.0731238769 
##            39            40            41            42            43 
## -0.2159883322  0.1906916777  0.0946876931 -0.2275679945 -0.1525371097 
##            44            45            46            47            48 
##  0.1300018414 -0.1147138365 -0.2589956260  0.1899850354 -0.3161668179 
##            49            50            51            52            53 
##  0.3867053934 -0.3345783381 -0.4369526860  0.2845604542 -0.0982301679 
##            54            55            56            57            58 
## -0.0576858931 -0.0343592112 -0.2016265071  0.3873933567 -0.4521257496 
##            59            60            61            62            63 
##  0.2030932678 -0.9996957016 -0.0834241173  0.4064736940 -0.6739386362 
##            64            65            66            67            68 
##  0.6500515278  0.2888967988 -0.1401042846  0.2437635033  0.3409057812 
##            69            70            71            72            73 
##  0.4372752538  0.4289103559  0.0699742487  0.1004894884  0.0385128060 
##            74            75            76            77            78 
##  0.3975788961  0.1445510860  0.4359685597  0.3961884747  0.1572905511 
##            79            80            81            82            83 
## -0.0688584248  0.0548405362 -0.0668721173 -0.3008902113  0.1551767082 
##            84            85            86            87            88 
##  0.0535327937  0.3137796690 -0.3563597833  0.3774734697 -0.0867984975 
##            89            90            91            92            93 
##  0.1422926133 -0.2301917063  0.0019893906  0.1178041550 -0.3152019523 
##            94            95            96            97            98 
##  0.1480502728 -0.0806682721  0.1346760781 -0.0075596558 -0.0467269548 
##            99           100           101           102           103 
## -0.0432559827 -0.0076412169  0.2168196555 -0.0567160085 -0.3390908528 
##           104           105           106           107           108 
##  0.4790871697 -0.5897972807  0.7242023645 -0.1390918266 -0.2205401444 
##           109           110           111           112           113 
##  0.2744621879 -0.2952138407  0.2346255303  0.3885728086 -0.2168426934 
##           114           115           116           117           118 
##  0.0704215965 -0.0577900675 -0.3424600300  0.5449024645 -0.4414096317 
##           119           120           121           122           123 
##  0.0139634066 -0.0911158315  0.0551179001 -0.2101234170  0.0432997754 
##           124           125           126           127           128 
## -0.4371741068  0.0670427222  0.2097003890 -0.4177036369 -0.5744809105 
##           129           130           131           132           133 
##  0.0003815003 -0.5590158326  0.3015835614 -0.0034151699  0.6444188985 
##           134           135           136           137           138 
## -0.1383251730 -0.0833493809 -0.2407875956 -0.1915845655  0.4579679402 
##           139           140           141           142           143 
## -0.5208660560  0.0105941921  0.4635803451 -0.3892899737 -0.5563893241 
##           144           145           146           147           148 
##  0.7413985628  0.2115761590 -0.1280963925  0.0360934534 -0.0845427242 
##           149           150           151           152           153 
## -0.1609383244  1.1769061139 -0.1982543588 -0.7362030298  0.0370950022 
##           154           155           156           157           158 
##  0.5195711821 -0.2564535584 -0.2508094121 -0.0089752939  0.2431248335 
##           159           160           161           162           163 
##  0.0598015411 -0.5597420171  0.3868125587  0.1882599644  0.1885316633 
##           164           165           166           167           168 
##  0.0716024647 -0.3930704232 -0.1934066136 -0.0066910866  0.4614248324 
##           169           170           171           172           173 
## -0.7131701187 -0.4709956717 -0.2870384853  0.5291480422 -0.3242413530 
##           174           175           176           177           178 
## -0.8298340449  0.4041531154  0.1276304965 -0.5429039118 -0.0354837656 
##           179           180           181           182           183 
## -0.3988197884  0.8453070169 -0.1335015813 -0.1538693089 -0.4670536196 
##           184           185           186           187           188 
##  0.1877092442 -0.2225580888  0.1971184331  0.3783298021  0.2219334015 
##           189           190           191           192           193 
##  0.2932752862  0.0331670835 -0.3470463393  0.3895564001 -0.1520519119 
##           194           195           196           197           198 
##  0.0915174536  0.1641764478 -0.0879766841  0.7055198031 -0.3422502154 
##           199           200           201           202           203 
##  0.1080211959  0.1811488507  0.1308019353  0.2672395797  0.1288288761 
##           204           205           206           207           208 
##  0.1382744024 -0.0531464969 -0.0817146247  0.2766920211  0.1733842202 
##           209           210           211           212           213 
## -0.0343331014  0.2132885591  0.1613767327 -0.0832999938 -0.2002098641 
##           214           215           216           217           218 
##  0.0013564061  0.1975033876 -0.0618602077 -0.1907173741  0.2085062466 
##           219           220           221           222           223 
## -0.0897268813 -0.0177740865  0.1848667304  0.0297133968  0.0364977979 
##           224           225           226           227           228 
## -0.2450449156  0.0562471803  0.0220350244  0.1721302586  0.0138078142 
##           229           230           231           232           233 
##  0.1464700409  0.1414996760  0.0171849435  0.1879138912  0.1723378711 
##           234           235           236           237           238 
## -0.2211668633 -0.0320712329  0.0535356726  0.2494976844 -0.0888111498 
##           239           240           241           242           243 
## -0.0374485099  0.1841046900 -0.1370826122 -0.2320307746  0.1751057456 
##           244           245           246           247           248 
## -0.0542607240 -0.1318471226  0.0301580398 -0.9338811605  0.7284126228 
##           249           250           251           252           253 
##  0.0452817700 -0.0630658937 -0.0541508147 -0.0208232305 -0.2441711683 
##           254           255           256           257           258 
##  0.1840758702 -0.2007153448 -0.3023033572  0.0469424852  0.0614292807 
##           259           260           261           262           263 
## -0.2256322917 -0.2097515374  0.0448562077 -0.1447953529 -0.2729797870 
##           264           265           266           267           268 
##  0.1529049706 -0.2221107797 -0.0350825744 -0.3195097615  0.1780991396 
##           269           270           271           272           273 
## -0.3017291283  0.4019499964 -0.1816576246  0.0818792616 -0.2094871128 
##           274           275           276           277           278 
## -0.2900234763  0.1307775203 -0.0403513956 -0.4708812581  0.0144410942 
##           279           280           281           282           283 
##  0.0770736744 -0.3412619609  0.0397136665  0.0302622391 -0.2344633746 
##           284           285           286           287           288 
##  0.1300016531 -0.1789750776  0.2057587317  0.0226660883 -0.1989084291 
##           289           290           291           292           293 
## -0.1120684412 -0.2397412151 -0.0392782358 -1.1099331151  0.0277465813 
##           294           295           296           297           298 
##  0.8134468042  0.1801553697 -0.4028874718 -0.0417753098  0.1777225426 
##           299           300           301           302           303 
## -0.4578844002  0.2797611481  0.2498422899 -0.3225890380  0.0973239160 
##           304           305           306           307           308 
## -0.0563959808 -0.5523187547  0.2775997568 -0.0728091678  0.1088573191 
##           309           310           311           312           313 
## -0.0143079123  0.0303605212 -0.2887662330  0.0850970535 -0.0329067302 
##           314           315           316           317           318 
##  0.0558424978  0.0714429960  0.2151174047 -0.2604565525  0.2927277071 
##           319           320           321           322           323 
## -0.0290093266  0.1033548556 -0.0557923009  0.2113095567 -0.0468579559 
##           324           325           326           327           328 
## -0.1037707187  0.3031576267 -0.0347421226 -0.4438926379  0.5354343401 
##           329           330           331           332           333 
##  0.8244056676 -0.2278170437  0.3064034411 -0.0950029865  0.3187677278 
##           334           335           336           337           338 
##  0.0364553992  0.5541284401 -0.2263816130 -0.0960241921  0.1735656633 
##           339           340           341           342           343 
## -0.1063674980  0.0007322949 -0.0836698713  0.0646073257  0.0468320615 
##           344           345           346           347           348 
## -0.0699086951  0.3492374117 -0.1399546655  0.2583635682  0.1635487302 
##           349           350           351           352           353 
##  0.2713485365  0.0947551267  0.2384196113  0.1607348651  0.4255646122 
##           354           355           356           357           358 
##  0.3234367625  0.3702992447 -0.0538621146 -0.5542460192  1.1612520366 
##           359           360           361           362           363 
##  0.6084843173  0.3741997602  0.3947697096  0.4210373576 -0.0069876449 
##           364           365           366           367           368 
##  0.0192161058  0.7048840448  0.5325366145  0.2638445158  0.3602227968 
##           369           370           371           372           373 
##  0.0073274245  0.0700647788 -0.1410292801  0.0012790709  0.5275607978 
##           374           375           376           377           378 
## -0.0807730962  0.1346814747  0.0464244512 -0.0141956828 -0.1709396754 
##           379           380           381           382           383 
## -1.6563794992  0.8131357271  0.7228473266  0.0689225947  0.0650020037 
##           384           385           386           387           388 
## -0.3095189300 -0.4470566349  0.2430711576  0.2046711042 -0.1559507123 
##           389           390           391           392           393 
## -0.0704081443 -0.1039765620 -0.1420811383 -0.2142016806  0.0439716024 
##           394           395           396           397           398 
##  0.2004573841 -0.3017913120  0.1466495081 -0.1248291692  0.0829192272 
##           399           400           401           402           403 
## -0.2029540430 -0.0880782667  0.3315544452 -0.0733011158  0.0748198866 
##           404           405           406           407           408 
##  0.1501675219  0.1106979804 -0.0703802965 -0.4734525118  0.4095880969 
##           409           410           411           412           413 
## -0.2157052210 -0.1303159789  0.0916831527  0.0650420951 -0.0104793383 
##           414           415           416           417           418 
##  0.1067086492 -0.1129985972  0.0395042732 -0.1965903626  0.0430635648 
##           419           420           421           422           423 
##  0.6149905011 -0.7140454885 -0.8736326222  0.6371071150 -0.3903998230 
##           424           425           426           427           428 
## -0.2519660828  0.2110009044 -0.4920854058 -0.5576390325  0.8774120716 
##           429           430           431           432           433 
## -0.7435744381 -0.3974246070 -0.2020093474 -0.8414679683  0.3659963034 
##           434           435           436           437           438 
##  0.0118324166 -0.4513501089 -0.5131203030  0.0295041091  0.8129727078 
##           439           440           441           442           443 
## -0.6287802858 -0.0929971713 -0.3083509108  0.1082324515  0.0988055485 
##           444           445           446           447           448 
## -0.3690935817  0.0660488377  0.1173975067 -0.4038086491 -0.4343779701 
##           449           450           451           452           453 
##  0.5969329346 -0.8859259106  1.0548088933 -0.1794915303 -0.3698129740 
##           454           455           456           457           458 
## -0.3768637563  0.0955181210 -0.0046434564  0.0878533483 -0.1332838731 
##           459           460           461           462           463 
##  0.0628778669  0.1489849237 -0.3798635980 -0.5143970495  0.5848923495 
##           464           465           466           467           468 
##  0.3394519417 -0.0728864082  0.1512234649  0.1860736199 -0.5618374466 
##           469           470           471           472           473 
##  0.0670099122 -0.7581023064  0.5528533862 -0.0874287966 -0.2056913222 
##           474           475           476           477           478 
##  0.0485617082 -0.4205897296 -0.3662484912  0.0624868292  0.3533179509 
##           479           480           481           482           483 
## -0.3642189220  0.0867524525 -0.2139411986 -0.0977374620  0.0211438736 
##           484           485           486           487           488 
## -0.0390661485  0.0663988680 -0.1460124344  0.0376205601 -0.0604986558 
##           489           490 
## -0.1240688686  0.1094587792

mod_ardl92_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                            data = full_cases_wastewater_weather_data_meck_train, 
                            p=2,q=9)
summary(mod_ardl92_meck)
## 
## Time series regression with "ts" data:
## Start = 10, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.76298 -0.16215  0.00894  0.17265  1.09150 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.740370   0.335063  -2.210 0.027613 *  
## log_viral_gene.t     -0.053319   0.035170  -1.516 0.130193    
## log_viral_gene.1      0.099892   0.045907   2.176 0.030057 *  
## log_viral_gene.2      0.003063   0.035356   0.087 0.931002    
## log_mean_new_cases.1  0.514385   0.046179  11.139  < 2e-16 ***
## log_mean_new_cases.2  0.072966   0.051631   1.413 0.158259    
## log_mean_new_cases.3  0.181462   0.051536   3.521 0.000472 ***
## log_mean_new_cases.4  0.081804   0.052151   1.569 0.117418    
## log_mean_new_cases.5  0.109375   0.051967   2.105 0.035851 *  
## log_mean_new_cases.6  0.074401   0.052030   1.430 0.153396    
## log_mean_new_cases.7 -0.006672   0.051322  -0.130 0.896621    
## log_mean_new_cases.8  0.003192   0.051478   0.062 0.950588    
## log_mean_new_cases.9 -0.103250   0.045770  -2.256 0.024542 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3133 on 468 degrees of freedom
## Multiple R-squared:  0.9217, Adjusted R-squared:  0.9197 
## F-statistic: 458.9 on 12 and 468 DF,  p-value: < 2.2e-16
f_ardl92_meck <- forecast(mod_ardl92_meck , 
                          x= t(full_cases_wastewater_weather_data_meck_test[,8]),
                          h=14, interval = TRUE)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl92_meck$forecasts[,2])
## [1] 0.1207945
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl92_meck$forecasts[,2])
## [1] 0.09736353
checkresiduals(mod_ardl92_meck)
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -3.092371e-02  1.163238e-02  1.252001e-02  1.014782e-01  2.836953e-02 
##            15            16            17            18            19 
##  2.833400e-02 -1.696518e-03 -1.227269e-01 -1.590525e-01 -3.045078e-02 
##            20            21            22            23            24 
##  2.184542e-01  5.078416e-02 -3.255922e-02 -1.214547e-01  2.311235e-01 
##            25            26            27            28            29 
##  4.023654e-02 -2.353277e-01 -8.977339e-02  1.961115e-01  5.112010e-03 
##            30            31            32            33            34 
## -3.149867e-02  1.427563e-02 -2.207462e-01 -2.851758e-02 -2.929289e-01 
##            35            36            37            38            39 
##  3.177582e-01  4.487638e-02 -9.499900e-02 -8.387231e-02 -2.265090e-01 
##            40            41            42            43            44 
##  8.259352e-02  9.959311e-02 -8.641751e-02 -1.764461e-01  1.229113e-01 
##            45            46            47            48            49 
## -7.265117e-02 -2.519822e-01  9.349147e-02 -2.707908e-01  3.195785e-01 
##            50            51            52            53            54 
## -2.050264e-01 -3.964405e-01  1.363230e-01  1.925282e-02  2.139667e-02 
##            55            56            57            58            59 
## -1.014455e-01 -1.146824e-01  3.385559e-01 -3.207601e-01  1.507240e-01 
##            60            61            62            63            64 
## -1.071141e+00 -2.682274e-01  2.178915e-01 -5.722560e-01  6.045407e-01 
##            65            66            67            68            69 
##  3.489794e-01  2.040119e-01 -3.519632e-02  2.839743e-01  2.117074e-01 
##            70            71            72            73            74 
##  3.177930e-01 -4.062772e-02 -6.585365e-02 -2.198761e-01  4.218497e-01 
##            75            76            77            78            79 
## -1.655438e-01  4.272526e-01  3.326260e-01  3.096256e-01 -1.568450e-02 
##            80            81            82            83            84 
## -6.525427e-02 -1.434785e-01 -3.334671e-01 -2.611591e-02 -4.666716e-02 
##            85            86            87            88            89 
##  3.474433e-01 -2.150304e-01  3.881112e-01  2.499551e-02  2.028181e-01 
##            90            91            92            93            94 
## -2.367724e-01 -8.491285e-02  1.206571e-02 -3.959261e-01  3.882358e-02 
##            95            96            97            98            99 
## -2.155568e-01  1.809715e-01  8.645893e-02  2.824313e-03 -1.786669e-02 
##           100           101           102           103           104 
## -8.591058e-02  2.274223e-01 -1.068923e-01 -3.708266e-01  3.465886e-01 
##           105           106           107           108           109 
## -4.702015e-01  6.073526e-01 -6.934208e-02 -1.380305e-01  1.090499e-01 
##           110           111           112           113           114 
## -3.081340e-01  9.294370e-02  2.802856e-01 -1.150670e-01 -4.287394e-02 
##           115           116           117           118           119 
## -1.069286e-01 -4.091438e-01  3.187158e-01 -3.036462e-01 -8.434482e-02 
##           120           121           122           123           124 
## -1.245349e-01  4.561678e-02 -1.409853e-01 -2.357035e-02 -3.462645e-01 
##           125           126           127           128           129 
## -4.734365e-03  3.363099e-01 -2.809807e-01 -6.658337e-01 -1.931057e-01 
##           130           131           132           133           134 
## -8.923531e-01  9.462148e-02 -1.521770e-01  7.699534e-01  7.831224e-02 
##           135           136           137           138           139 
##  1.447998e-01 -2.591512e-01 -2.804563e-01  1.452160e-01 -5.340732e-01 
##           140           141           142           143           144 
## -1.774752e-01  3.831458e-01 -2.355669e-01 -5.379080e-01  4.430436e-01 
##           145           146           147           148           149 
##  4.268551e-01  7.313508e-02  8.329347e-02 -2.276817e-01 -3.157568e-01 
##           150           151           152           153           154 
##  9.449980e-01 -1.127343e-02 -7.474956e-01 -3.350319e-01  6.629206e-01 
##           155           156           157           158           159 
## -3.926999e-01 -2.283320e-01 -2.171595e-01  3.702117e-01  2.552329e-01 
##           160           161           162           163           164 
## -3.746085e-01  3.234608e-01  3.548603e-01  3.353210e-01  1.897544e-01 
##           165           166           167           168           169 
## -3.546162e-01 -2.689742e-01 -1.656706e-01  3.926646e-01 -5.638034e-01 
##           170           171           172           173           174 
## -6.236650e-01 -3.644881e-01  2.419904e-01  6.867030e-02 -7.785484e-01 
##           175           176           177           178           179 
##  3.596520e-01  2.556221e-01 -2.183095e-01  5.990763e-02 -4.009686e-01 
##           180           181           182           183           184 
##  9.059714e-01  2.544917e-01  1.393752e-01 -4.475876e-01  5.457393e-02 
##           185           186           187           188           189 
## -2.374567e-01  8.340786e-02  4.492369e-01  4.403853e-01  5.910953e-01 
##           190           191           192           193           194 
##  2.577866e-01 -1.954837e-01  1.473030e-01 -1.985440e-01 -4.717432e-02 
##           195           196           197           198           199 
##  7.214516e-02 -1.357494e-02  7.170069e-01 -1.940442e-02  1.830457e-01 
##           200           201           202           203           204 
##  1.607166e-01  1.394537e-01  2.803610e-01  1.726479e-01  1.823049e-01 
##           205           206           207           208           209 
##  3.561684e-02 -9.836023e-02  1.927115e-01  6.261834e-02 -1.391065e-02 
##           210           211           212           213           214 
##  1.735067e-01  1.859842e-01  7.134149e-02 -1.744462e-01 -5.452612e-02 
##           215           216           217           218           219 
##  1.558357e-01 -3.467078e-02 -1.510130e-01  1.601537e-01 -2.934839e-02 
##           220           221           222           223           224 
##  5.364477e-02  1.838195e-01  3.012772e-02  9.895059e-02 -2.495102e-01 
##           225           226           227           228           229 
## -1.737132e-02 -1.475369e-02  1.207811e-01  8.583218e-02  1.213871e-01 
##           230           231           232           233           234 
##  1.855354e-01  4.753981e-02  1.866039e-01  1.354519e-01 -1.621541e-01 
##           235           236           237           238           239 
## -1.476087e-01 -9.523335e-03  1.469191e-01 -5.875478e-02 -6.073574e-02 
##           240           241           242           243           244 
##  1.394707e-01 -9.513615e-02 -2.445320e-01  8.757945e-02 -2.934725e-02 
##           245           246           247           248           249 
## -9.233432e-02  2.892464e-02 -8.366786e-01  5.078695e-01  2.107009e-01 
##           250           251           252           253           254 
##  2.047745e-01  8.030800e-02  1.051612e-01 -1.453955e-01  9.064731e-02 
##           255           256           257           258           259 
## -1.313186e-01 -4.096315e-01 -1.006922e-02  7.446587e-02 -1.236577e-01 
##           260           261           262           263           264 
## -1.928842e-01  6.074715e-02 -1.290528e-01 -2.034470e-01  7.606261e-02 
##           265           266           267           268           269 
## -1.389784e-01 -2.766717e-02 -2.661200e-01  1.163460e-01 -2.396346e-01 
##           270           271           272           273           274 
##  4.097262e-01 -4.466364e-02  1.329093e-01 -1.376155e-01 -3.050477e-01 
##           275           276           277           278           279 
##  6.348342e-02 -1.009835e-01 -3.900220e-01 -1.174700e-01  1.093365e-01 
##           280           281           282           283           284 
## -2.734615e-01  4.514930e-02 -1.382426e-02 -1.349219e-01  1.037062e-01 
##           285           286           287           288           289 
## -1.236246e-01  1.722392e-01  6.008186e-02 -1.184771e-01 -1.498106e-01 
##           290           291           292           293           294 
## -3.679851e-01 -1.146621e-01 -1.211610e+00 -2.829368e-01  6.819258e-01 
##           295           296           297           298           299 
##  5.833887e-01  5.790571e-02  1.464892e-01  3.321741e-01 -3.720783e-01 
##           300           301           302           303           304 
##  1.564222e-01  1.168640e-01 -2.581056e-01  1.968431e-02  2.939600e-02 
##           305           306           307           308           309 
## -5.594626e-01  2.314759e-01 -7.625572e-02  1.770047e-01  6.695224e-02 
##           310           311           312           313           314 
##  1.017863e-01 -2.388097e-01  2.251281e-03 -3.007638e-02  6.243413e-02 
##           315           316           317           318           319 
##  1.010172e-01  2.798638e-01 -1.123650e-01  1.994614e-01  2.516946e-02 
##           320           321           322           323           324 
##  1.304755e-02 -9.812070e-02  1.101769e-01 -5.688409e-02 -1.611528e-01 
##           325           326           327           328           329 
##  2.100385e-01 -4.400001e-02 -5.062325e-01  2.867265e-01  8.064437e-01 
##           330           331           332           333           334 
##  3.036679e-02  3.228944e-01 -6.375375e-02  2.207628e-01  3.847121e-02 
##           335           336           337           338           339 
##  4.157262e-01 -1.862650e-01 -2.152798e-01  1.450628e-01 -1.573700e-01 
##           340           341           342           343           344 
## -3.183065e-02 -9.206769e-02  1.181346e-01  7.614366e-02  2.500905e-02 
##           345           346           347           348           349 
##  3.277367e-01  7.651524e-03  2.772196e-01  2.234536e-01  3.496113e-01 
##           350           351           352           353           354 
##  1.703349e-01  2.441747e-01  2.443558e-01  3.307703e-01  4.035216e-01 
##           355           356           357           358           359 
##  3.452711e-01 -2.891394e-02 -7.149718e-01  7.580044e-01  6.009496e-01 
##           360           361           362           363           364 
##  5.801873e-01  4.424346e-01  4.607344e-01 -7.361956e-02 -2.887587e-01 
##           365           366           367           368           369 
##  2.649712e-01  2.405537e-01  2.796954e-01  2.958525e-01  2.132516e-01 
##           370           371           372           373           374 
## -2.750058e-02 -2.824429e-01 -1.764350e-01  2.016469e-01 -1.351666e-02 
##           375           376           377           378           379 
##  1.020631e-01  6.526443e-02  1.155335e-01 -2.553103e-01 -1.762980e+00 
##           380           381           382           383           384 
##  2.101670e-01  5.359992e-01  4.782681e-01  3.562092e-01 -8.033185e-02 
##           385           386           387           388           389 
## -3.883561e-01  2.627346e-02  7.952609e-02 -2.607643e-01 -1.175277e-01 
##           390           391           392           393           394 
## -5.857509e-02 -1.150789e-01 -2.598925e-01 -4.213645e-02  1.198930e-01 
##           395           396           397           398           399 
## -2.661433e-01  1.489386e-01 -2.024840e-01  7.704819e-02 -2.903232e-01 
##           400           401           402           403           404 
## -1.647626e-01  1.643158e-01 -1.258556e-01  7.388809e-02 -7.547620e-02 
##           405           406           407           408           409 
##  4.738201e-02 -1.695564e-01 -5.873658e-01  1.816129e-01 -3.658345e-01 
##           410           411           412           413           414 
## -1.541965e-01 -1.419844e-01  8.592332e-03 -7.089878e-03  1.165630e-01 
##           415           416           417           418           419 
## -6.272235e-02 -5.778883e-02 -2.247626e-01 -2.778469e-01  4.572784e-01 
##           420           421           422           423           424 
## -6.440206e-01 -1.030302e+00  3.249665e-01 -4.085808e-01 -1.785736e-01 
##           425           426           427           428           429 
##  7.035872e-02 -3.651298e-01 -5.662069e-01  7.933071e-01 -4.768814e-01 
##           430           431           432           433           434 
## -2.922878e-01 -2.685913e-01 -7.243048e-01  1.725235e-01  1.543828e-01 
##           435           436           437           438           439 
## -1.620727e-01 -4.584507e-01  5.191641e-02  9.345671e-01 -3.536851e-01 
##           440           441           442           443           444 
##  6.352328e-02 -3.436279e-01  1.305824e-01  2.842651e-01 -3.008418e-01 
##           445           446           447           448           449 
##  1.386451e-01  1.215209e-01 -2.017925e-01 -4.699615e-01  4.751846e-01 
##           450           451           452           453           454 
## -7.258056e-01  1.091502e+00  1.442230e-01 -1.525797e-02 -4.045037e-01 
##           455           456           457           458           459 
##  1.532905e-02 -3.790277e-02  8.403449e-02 -3.366523e-02  2.058621e-01 
##           460           461           462           463           464 
##  2.752573e-01 -1.232036e-01 -5.652366e-01  4.062207e-01  4.750853e-01 
##           465           466           467           468           469 
##  1.755680e-01  3.300130e-01  3.608029e-01 -3.666815e-01 -5.185949e-02 
##           470           471           472           473           474 
## -6.779433e-01  3.216355e-01  7.937263e-02  5.339369e-02  1.819193e-01 
##           475           476           477           478           479 
## -1.874273e-01 -3.059841e-01  8.937394e-03  4.802756e-01 -1.195863e-01 
##           480           481           482           483           484 
##  2.334788e-01 -2.064050e-02 -4.314131e-02  6.413132e-02 -4.859655e-03 
##           485           486           487           488           489 
##  1.579740e-01 -5.826499e-02  1.890077e-01  6.867093e-02 -4.940313e-05 
##           490 
##  1.385353e-01

exp(f_ardl92_meck $forecasts[1,2])
## [1] 3.694923
exp(f_ardl92_meck $forecasts[1,1])
## [1] 1.990954
exp(f_ardl92_meck $forecasts[1,3])
## [1] 6.8191
exp(f_ardl92_meck $forecasts[1,2]) - exp(full_cases_wastewater_weather_data_meck_test[1,7])
## [1] 0.668796
exp(f_ardl92_meck $forecasts[7,2])
## [1] 3.803504
exp(f_ardl92_meck $forecasts[7,1])
## [1] 1.601884
exp(f_ardl92_meck $forecasts[7,3])
## [1] 8.752501
exp(f_ardl92_meck $forecasts[7,2]) - exp(full_cases_wastewater_weather_data_meck_test[7,7])
## [1] -0.05095394
exp(f_ardl92_meck $forecasts[14,2])
## [1] 3.852675
exp(f_ardl92_meck $forecasts[14,1])
## [1] 1.268613
exp(f_ardl92_meck $forecasts[14,3])
## [1] 13.59007
exp(f_ardl92_meck $forecasts[14,2]) - exp(full_cases_wastewater_weather_data_meck_test[14,7])
## [1] 0.8327343
lowest_rmse_weather_meck <- Inf
best_mod_weather_meck <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    remove = list(p =list(TAVG=c(1:p),mean_precipation=c(1:p)))
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                     TAVG,data = full_cases_wastewater_weather_data_meck_train, 
                   p=p,q=q,
                   remove = remove)
    f <- forecast(mod, x= t(full_cases_wastewater_weather_data_meck_test[,c(8,5,6)]),h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
                         f$forecasts) #interchanged between RMSE and MAE 
    if (forecast_acc<lowest_rmse_weather_meck){
      lowest_rmse_weather_meck <- forecast_acc
      best_mod_weather_meck <- mod 
    }
  }
}

lowest_rmse_weather_meck #0.118
## [1] 0.1178899
summary(best_mod_weather_meck) #ARDL(8,13)(lowest RMSE), ARDL(8,14)(lowest MAE)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.69824 -0.16711  0.00604  0.17618  1.01867 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.125372   0.478060  -0.262 0.793247    
## log_viral_gene.t     -0.063206   0.035437  -1.784 0.075160 .  
## log_viral_gene.1      0.125407   0.046027   2.725 0.006687 ** 
## log_viral_gene.2     -0.050328   0.046166  -1.090 0.276229    
## log_viral_gene.3      0.116435   0.046220   2.519 0.012109 *  
## log_viral_gene.4     -0.111584   0.046575  -2.396 0.016990 *  
## log_viral_gene.5      0.067784   0.046795   1.449 0.148160    
## log_viral_gene.6     -0.068752   0.046795  -1.469 0.142468    
## log_viral_gene.7      0.044782   0.046578   0.961 0.336846    
## log_viral_gene.8     -0.022033   0.046471  -0.474 0.635645    
## log_viral_gene.9      0.007141   0.046057   0.155 0.876852    
## log_viral_gene.10     0.006222   0.045832   0.136 0.892077    
## log_viral_gene.11     0.039933   0.045799   0.872 0.383719    
## log_viral_gene.12    -0.059399   0.045111  -1.317 0.188601    
## log_viral_gene.13    -0.030162   0.034517  -0.874 0.382681    
## mean_precipation.t   -0.091274   0.058839  -1.551 0.121542    
## TAVG.t                0.001810   0.001145   1.581 0.114588    
## log_mean_new_cases.1  0.515103   0.047083  10.940  < 2e-16 ***
## log_mean_new_cases.2  0.044340   0.052916   0.838 0.402512    
## log_mean_new_cases.3  0.202523   0.052882   3.830 0.000146 ***
## log_mean_new_cases.4  0.046269   0.053482   0.865 0.387430    
## log_mean_new_cases.5  0.135124   0.053921   2.506 0.012563 *  
## log_mean_new_cases.6  0.061860   0.052991   1.167 0.243676    
## log_mean_new_cases.7  0.004953   0.052940   0.094 0.925506    
## log_mean_new_cases.8 -0.029338   0.048120  -0.610 0.542382    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3125 on 452 degrees of freedom
## Multiple R-squared:  0.9241, Adjusted R-squared:  0.9201 
## F-statistic: 229.3 on 24 and 452 DF,  p-value: < 2.2e-16
remove <- list(p =list(TAVG=c(1:13),mean_precipation=c(1:13)))
mod_ardl813_weather_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene + 
                                      mean_precipation +
                                      TAVG,
                                    data = full_cases_wastewater_weather_data_meck_train,
                                    p=13,q=8,
                                    remove = remove)
f_ardl813_weather_meck <- forecast(mod_ardl813_weather_meck, 
                                   x= t(full_cases_wastewater_weather_data_meck_test[,c(8,5,6)]),
                                   h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl813_weather_meck$forecasts) 
## [1] 0.1178899
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl813_weather_meck$forecasts) 
## [1] 0.09683786
checkresiduals(mod_ardl813_weather_meck)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
## -0.0203143239  0.0642146685  0.0221526864 -0.0962701813 -0.1417690848 
##            19            20            21            22            23 
## -0.0157338728  0.2341887783  0.0319256507 -0.0052776122 -0.0456536696 
##            24            25            26            27            28 
##  0.2605503730  0.0744736679 -0.2200575064 -0.0301138716  0.3005537191 
##            29            30            31            32            33 
##  0.0502998395 -0.0182677882  0.0108076095 -0.1362496950 -0.0632185773 
##            34            35            36            37            38 
## -0.3611747982  0.4360537756  0.0443800585 -0.0830318601 -0.1249826246 
##            39            40            41            42            43 
## -0.2313653802  0.1678936839  0.1490431467 -0.0997737971 -0.1034627890 
##            44            45            46            47            48 
##  0.2040485017 -0.0409551952 -0.1914122679  0.1739588227 -0.2433583639 
##            49            50            51            52            53 
##  0.4119306818 -0.1800399409 -0.3258329159  0.1656971234 -0.0269440642 
##            54            55            56            57            58 
##  0.0174023405  0.0570945322 -0.1232945650  0.3665669388 -0.3215899330 
##            59            60            61            62            63 
##  0.1705313106 -1.0076204852 -0.2970204584  0.1920336460 -0.4654790854 
##            64            65            66            67            68 
##  0.5966773420  0.4124931965  0.0928354364  0.1501158637  0.2192061455 
##            69            70            71            72            73 
##  0.3329158026  0.2723225144 -0.0163482285  0.0601135859 -0.0757139970 
##            74            75            76            77            78 
##  0.2047617535  0.0682958197  0.3640765833  0.3212311459  0.1721120298 
##            79            80            81            82            83 
## -0.1584374838 -0.1519486293 -0.2896536998 -0.4290702151 -0.1903976126 
##            84            85            86            87            88 
## -0.0048197007  0.3464351330 -0.2376521993  0.3583399143  0.0169329525 
##            89            90            91            92            93 
##  0.2159660604 -0.2381066654 -0.0613369296 -0.0092211788 -0.3701270765 
##            94            95            96            97            98 
## -0.0359904190 -0.1652836823  0.1084614693 -0.0002161939  0.0211436694 
##            99           100           101           102           103 
## -0.0591298835 -0.0540648466  0.1511041317 -0.0503977813 -0.3401795942 
##           104           105           106           107           108 
##  0.3362220203 -0.5849450853  0.5720154100 -0.0841694877 -0.1257906198 
##           109           110           111           112           113 
##  0.1472849240 -0.2908293039  0.1002674005  0.3246578791 -0.1556118761 
##           114           115           116           117           118 
## -0.0232181809 -0.1783368264 -0.4536467881  0.3043025317 -0.4324920378 
##           119           120           121           122           123 
## -0.0599007571 -0.2124356139  0.0724487815 -0.2040099066  0.0068871325 
##           124           125           126           127           128 
## -0.4251489233  0.0159808478  0.1693657510 -0.2771325888 -0.6149342534 
##           129           130           131           132           133 
## -0.2489195194 -0.6951349156  0.0378558678 -0.1040001309  0.6821632225 
##           134           135           136           137           138 
##  0.1007378456  0.0828216046 -0.2662099146 -0.3048662113  0.2331487310 
##           139           140           141           142           143 
## -0.5227913064 -0.1700500211  0.2930506640 -0.3327859785 -0.6874968738 
##           144           145           146           147           148 
##  0.4137984189  0.2588582952  0.0112010002 -0.0129364056 -0.1308384686 
##           149           150           151           152           153 
## -0.2848042777  0.9031607272 -0.0439192902 -0.6190688700 -0.3086649420 
##           154           155           156           157           158 
##  0.2384689007 -0.1914791623 -0.2242573370 -0.1310967675  0.2172074459 
##           159           160           161           162           163 
##  0.1148629572 -0.5143582486  0.2765788871  0.2269333201  0.3186197457 
##           164           165           166           167           168 
##  0.2095242376 -0.3321430121 -0.3050401549 -0.2157816192  0.4431785682 
##           169           170           171           172           173 
## -0.5847592376 -0.5584305108 -0.5088650127  0.4110744806 -0.1861459192 
##           174           175           176           177           178 
## -0.7199975478  0.2252349236  0.2517954784 -0.3189681924 -0.0549944957 
##           179           180           181           182           183 
## -0.2757485545  0.9648990680  0.2407087603  0.0831697982 -0.3769729565 
##           184           185           186           187           188 
##  0.1382628159 -0.2492918404  0.1696000020  0.3851359916  0.4847679801 
##           189           190           191           192           193 
##  0.5096161033  0.2730054054 -0.2306503505  0.2431031331 -0.1628124671 
##           194           195           196           197           198 
## -0.0354470449  0.0377364501 -0.0529059984  0.8359939803 -0.1237935272 
##           199           200           201           202           203 
##  0.1166828888  0.0761733649  0.1491359308  0.2271275329  0.1647063447 
##           204           205           206           207           208 
##  0.1937643184  0.0297349997 -0.1768733680  0.0876970550  0.0980910344 
##           209           210           211           212           213 
## -0.0567716459  0.1487311684  0.1579953238 -0.0366468213 -0.2507853419 
##           214           215           216           217           218 
## -0.1329910814  0.1357272398 -0.0340847439 -0.1860000523  0.1185787079 
##           219           220           221           222           223 
## -0.0749696700 -0.0364475867  0.1233545133  0.0562435556  0.0464463407 
##           224           225           226           227           228 
## -0.2277725342  0.0256302342 -0.0131479983  0.1915516828  0.0069905379 
##           229           230           231           232           233 
##  0.1589004130  0.1468468861  0.0609039534  0.0988501618  0.1139518272 
##           234           235           236           237           238 
## -0.2555130367 -0.1765469280 -0.1011533562  0.1451295796 -0.1170498461 
##           239           240           241           242           243 
## -0.1069472144  0.0937285546 -0.1042433936 -0.2428256709  0.0084470621 
##           244           245           246           247           248 
## -0.0887202380 -0.1354281689 -0.0008655397 -0.9009830281  0.5155341314 
##           249           250           251           252           253 
##  0.1302273662  0.1935987033  0.0364842877  0.1153657797 -0.1826642620 
##           254           255           256           257           258 
##  0.1166954798 -0.2152168583 -0.2977803105 -0.0490304892  0.0606004134 
##           259           260           261           262           263 
## -0.1525590628 -0.1978753728  0.0372361360 -0.0457467040 -0.1901918886 
##           264           265           266           267           268 
##  0.1060376845 -0.1454021675  0.0035783867 -0.3152010995  0.1425978952 
##           269           270           271           272           273 
## -0.2661500917  0.4020322085 -0.0713545879  0.1968470126 -0.1604115039 
##           274           275           276           277           278 
## -0.2802755530  0.0060378996 -0.0484880253 -0.4291855829 -0.0750472865 
##           279           280           281           282           283 
##  0.0603642707 -0.2694157213 -0.0220139788  0.0121326400 -0.1671129219 
##           284           285           286           287           288 
##  0.0815255705 -0.1617992175  0.2065798945  0.0945147066 -0.1007847523 
##           289           290           291           292           293 
## -0.1355236781 -0.2935207428 -0.1463894937 -1.2190918347 -0.3122416813 
##           294           295           296           297           298 
##  0.6476767936  0.5440209219 -0.0006621298  0.1384734125  0.3152930534 
##           299           300           301           302           303 
## -0.2814223710  0.1573515046  0.2357901423 -0.1613190481  0.0817917999 
##           304           305           306           307           308 
## -0.0245059674 -0.4605745667  0.1736199335 -0.0429403702  0.2105257434 
##           309           310           311           312           313 
##  0.0746441784  0.1449202538 -0.2215348207  0.0336543743 -0.0588420883 
##           314           315           316           317           318 
##  0.0596107572  0.1031129090  0.3498509593 -0.1064313038  0.2899534055 
##           319           320           321           322           323 
## -0.0205208657  0.1018861166 -0.0906392809  0.1761752557 -0.0465052925 
##           324           325           326           327           328 
## -0.1328821294  0.1842169998 -0.0432170205 -0.4770804196  0.3125547085 
##           329           330           331           332           333 
##  0.8188515834  0.0363209395  0.3361401139 -0.0938030912  0.2653813301 
##           334           335           336           337           338 
## -0.0907210149  0.4405444471 -0.1965091300 -0.1476269257  0.0022000190 
##           339           340           341           342           343 
## -0.0994442345 -0.0734120983 -0.1400562709  0.0203673945  0.0837196017 
##           344           345           346           347           348 
##  0.0061852548  0.3650879241 -0.0124002444  0.2974099695  0.1955461851 
##           349           350           351           352           353 
##  0.3657893799  0.2458340245  0.3719393571  0.2437855614  0.4733702273 
##           354           355           356           357           358 
##  0.3900479357  0.4198671850 -0.0546503866 -0.7072608675  0.7241529505 
##           359           360           361           362           363 
##  0.5605970772  0.5083266504  0.4049443815  0.4553483873 -0.1007641098 
##           364           365           366           367           368 
## -0.2922083740  0.3851627220  0.3748535489  0.2250282519  0.2944493854 
##           369           370           371           372           373 
##  0.0008209392 -0.0409349563 -0.3764073751 -0.2435793033  0.2704995089 
##           374           375           376           377           378 
## -0.0940332748  0.0850883489 -0.0355860857 -0.0496652599 -0.2423150631 
##           379           380           381           382           383 
## -1.6982410726  0.2147201745  0.5910289662  0.5142618292  0.3592949699 
##           384           385           386           387           388 
##  0.0261799756 -0.3438853076  0.0164080542  0.0618316204 -0.1035605100 
##           389           390           391           392           393 
## -0.0344583807 -0.0632393354 -0.1002096507 -0.2454546772 -0.0320204323 
##           394           395           396           397           398 
##  0.1978751642 -0.1961750837  0.1356950772 -0.1139990671  0.0841407471 
##           399           400           401           402           403 
## -0.2239419457 -0.1305148047  0.2403128705 -0.0561712495  0.0080853520 
##           404           405           406           407           408 
##  0.0354909578  0.0599148370 -0.1226677317 -0.6015776911  0.1255111531 
##           409           410           411           412           413 
## -0.3329367310 -0.2666023176 -0.1096017175  0.0346233057 -0.0057395080 
##           414           415           416           417           418 
##  0.0443203671 -0.1754897024 -0.0780696711 -0.3307083468 -0.1517845636 
##           419           420           421           422           423 
##  0.4567044639 -0.5710228308 -0.9909538629  0.1914160273 -0.4169079328 
##           424           425           426           427           428 
## -0.3053929428  0.0819314030 -0.3282032228 -0.5533628793  0.6174301855 
##           429           430           431           432           433 
## -0.5033940628 -0.1625260107 -0.2903992608 -0.7133682638  0.3616917890 
##           434           435           436           437           438 
##  0.1832756511 -0.1103716949 -0.3624778946  0.0730063877  0.9683172477 
##           439           440           441           442           443 
## -0.2617572985  0.0338017321 -0.2414802379  0.2411562596  0.1829585497 
##           444           445           446           447           448 
## -0.1895947232  0.1554974354  0.2711745745 -0.1850795987 -0.3982683933 
##           449           450           451           452           453 
##  0.4747383315 -0.6764453710  1.0186715030  0.0903838056  0.1060215837 
##           454           455           456           457           458 
## -0.3651305119  0.0647948187  0.0057320166  0.1768641626  0.1081144227 
##           459           460           461           462           463 
##  0.2303513065  0.2961644533 -0.1866256958 -0.4950846855  0.4418629718 
##           464           465           466           467           468 
##  0.4809856021  0.2157767960  0.2845607589  0.3800488446 -0.3451205740 
##           469           470           471           472           473 
##  0.0324299880 -0.6738465247  0.5175620400  0.0354623163  0.0692399292 
##           474           475           476           477           478 
##  0.1709385618 -0.2035540862 -0.3212426396 -0.0260255152  0.4063769225 
##           479           480           481           482           483 
## -0.0812750750  0.3082155777 -0.0068808216  0.0722527342  0.0773470752 
##           484           485           486           487           488 
##  0.0622704061  0.1864157543  0.0290805001  0.1879467929  0.0630865018 
##           489           490 
##  0.0070184936  0.1902128584

remove <- list(p =list(TAVG=c(1:14),mean_precipation=c(1:14)))
mod_ardl814_weather_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene + 
                                      mean_precipation +
                                      TAVG,
                                    data = full_cases_wastewater_weather_data_meck_train, 
                                    p=14,q=8,
                                    remove = remove)
f_ardl814_weather_meck <- forecast(mod_ardl814_weather_meck, 
                                   x= t(full_cases_wastewater_weather_data_meck_test[,c(8,5,6)]),
                                   h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl814_weather_meck$forecasts) 
## [1] 0.1185831
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl814_weather_meck$forecasts) 
## [1] 0.09460726
checkresiduals(mod_ardl814_weather_meck)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
##  0.0236284941  0.0441732705 -0.1013308735 -0.1415546485 -0.0188112659 
##            20            21            22            23            24 
##  0.2342149885  0.0030395560  0.0220448667 -0.0538295688  0.2568804434 
##            25            26            27            28            29 
##  0.0704903957 -0.2252088926 -0.0289202885  0.3049585789  0.0520915087 
##            30            31            32            33            34 
## -0.0382491677  0.0128643359 -0.1372025628 -0.0664549159 -0.3567931199 
##            35            36            37            38            39 
##  0.4476225861  0.0497800708 -0.1005062715 -0.1270770989 -0.2315188594 
##            40            41            42            43            44 
##  0.1635325890  0.1477529951 -0.1325181590 -0.0946654146  0.2428031801 
##            45            46            47            48            49 
## -0.0417902607 -0.1964889486  0.1649116168 -0.2408897241  0.4287377655 
##            50            51            52            53            54 
## -0.1831911204 -0.3467905165  0.1629964958 -0.0267752040  0.0118260240 
##            55            56            57            58            59 
##  0.0494214356 -0.1235984436  0.3499110306 -0.3199827675  0.1682726052 
##            60            61            62            63            64 
## -1.0085887487 -0.2999942953  0.1997490453 -0.4318964866  0.5990048053 
##            65            66            67            68            69 
##  0.3815739309  0.0919405600  0.1422469696  0.2136003566  0.3302354541 
##            70            71            72            73            74 
##  0.2443307912 -0.0146388136  0.0635284063 -0.0879316563  0.2067473834 
##            75            76            77            78            79 
##  0.0594774874  0.3658710879  0.2391088805  0.1735827042 -0.1554051567 
##            80            81            82            83            84 
## -0.1322345524 -0.2721308779 -0.4239276762 -0.1623063923  0.1321196864 
##            85            86            87            88            89 
##  0.3548218837 -0.2333162531  0.3570649839  0.0158628806  0.2131095069 
##            90            91            92            93            94 
## -0.2362302468 -0.0514782308  0.0001655113 -0.3569912686 -0.0336831873 
##            95            96            97            98            99 
## -0.1651957691  0.1069405837  0.0031111717  0.0318995578 -0.0563610955 
##           100           101           102           103           104 
## -0.0650883051  0.1491959798 -0.0533842550 -0.3421583805  0.3376886883 
##           105           106           107           108           109 
## -0.6183869002  0.5842914235 -0.0382588630 -0.1300022410  0.1472675420 
##           110           111           112           113           114 
## -0.2939161931  0.1006226223  0.3008592344 -0.1466933100 -0.0131525750 
##           115           116           117           118           119 
## -0.1783547371 -0.4541350703  0.3035087539 -0.4268096464 -0.0644256607 
##           120           121           122           123           124 
## -0.2018956226  0.0667072564 -0.2051468384  0.0066737258 -0.4222166081 
##           125           126           127           128           129 
##  0.0207517385  0.1647533938 -0.2685480695 -0.5684281897 -0.2477791594 
##           130           131           132           133           134 
## -0.6933166002  0.0419610590 -0.1002127191  0.6599853944  0.1188468921 
##           135           136           137           138           139 
##  0.0808087262 -0.2801234659 -0.3104294378  0.2240874271 -0.5253082948 
##           140           141           142           143           144 
## -0.2378945090  0.3108061672 -0.3482106881 -0.6757198201  0.4206199928 
##           145           146           147           148           149 
##  0.2696302549  0.0243925865  0.0373229073 -0.1249584925 -0.2707408926 
##           150           151           152           153           154 
##  0.9020316796 -0.0484097728 -0.6301381852 -0.3037500687  0.1939051426 
##           155           156           157           158           159 
## -0.1845633041 -0.1763075348 -0.1213200304  0.1758149268  0.1179527149 
##           160           161           162           163           164 
## -0.5001809725  0.2918538518  0.2435255328  0.3258668327  0.3071738504 
##           165           166           167           168           169 
## -0.3933762130 -0.3049529355 -0.2113146384  0.4748776811 -0.5787017938 
##           170           171           172           173           174 
## -0.5547854749 -0.5075730040  0.4086640235 -0.1796712673 -0.7205060184 
##           175           176           177           178           179 
##  0.2201998398  0.2673079548 -0.3197223815 -0.0624276323 -0.2377404135 
##           180           181           182           183           184 
##  0.9699082138  0.2263955101  0.0092263345 -0.3163487297  0.1305194456 
##           185           186           187           188           189 
## -0.2540554785  0.1654559519  0.3526391356  0.4858152825  0.5139030740 
##           190           191           192           193           194 
##  0.3060530405 -0.2424893311  0.2350329795 -0.1709678617 -0.0567108136 
##           195           196           197           198           199 
##  0.0449879304 -0.0450563462  0.8330783226 -0.1258464551  0.1165182864 
##           200           201           202           203           204 
##  0.0813614631  0.1794064915  0.2281824148  0.1657279687  0.1975605229 
##           205           206           207           208           209 
##  0.0230460542 -0.1760630547  0.0929005653  0.1326454144 -0.0557525528 
##           210           211           212           213           214 
##  0.1499882354  0.1480563979 -0.0377810282 -0.2526197942 -0.1301589753 
##           215           216           217           218           219 
##  0.1774252961 -0.0360362370 -0.1841879666  0.1020190037 -0.0755621113 
##           220           221           222           223           224 
## -0.0368125592  0.1237322682  0.0871143151  0.0427377765 -0.2319583814 
##           225           226           227           228           229 
##  0.0319141591 -0.0222694121  0.1816045460  0.0013253019  0.1400025279 
##           230           231           232           233           234 
##  0.1446182557  0.0567527435  0.0679309681  0.1141727436 -0.2579486863 
##           235           236           237           238           239 
## -0.1773018223 -0.0748104964  0.1479071826 -0.1142701971 -0.1238770312 
##           240           241           242           243           244 
##  0.0922216719 -0.1099720780 -0.2495378726  0.0010096046 -0.0854946440 
##           245           246           247           248           249 
## -0.1297934119  0.0212590987 -0.9053922935  0.5109879508  0.1285605489 
##           250           251           252           253           254 
##  0.1836367794  0.0323123867  0.1141870341 -0.1803387958  0.1072485384 
##           255           256           257           258           259 
## -0.2209768567 -0.3017163231 -0.0375685420  0.0570896392 -0.1538909235 
##           260           261           262           263           264 
## -0.2009558465  0.0293518255 -0.0561042367 -0.2010049478  0.0864297269 
##           265           266           267           268           269 
## -0.1473870790 -0.0022179643 -0.3155888998  0.1359295400 -0.2700443246 
##           270           271           272           273           274 
##  0.3959021332 -0.0712077442  0.1883091508 -0.1654777886 -0.3081536581 
##           275           276           277           278           279 
##  0.0007089265 -0.0531822471 -0.4335907088 -0.0913175170  0.0613128537 
##           280           281           282           283           284 
## -0.2642509784 -0.0263912153  0.0086749298 -0.1681298841  0.0788645541 
##           285           286           287           288           289 
## -0.1524532506  0.2020784779  0.0937343444 -0.1162750269 -0.1416532676 
##           290           291           292           293           294 
## -0.2982481644 -0.1496078734 -1.2524779736 -0.3048938154  0.6609610801 
##           295           296           297           298           299 
##  0.5397187457 -0.0075262312  0.1320070578  0.3108173288 -0.2799082907 
##           300           301           302           303           304 
##  0.1512174843  0.2406220204 -0.1673251286  0.0773874138 -0.0287314645 
##           305           306           307           308           309 
## -0.4632487709  0.1786821601 -0.0416407075  0.2186550104  0.0727769785 
##           310           311           312           313           314 
##  0.1407108662 -0.2248459064  0.0323964131 -0.0888312646  0.0647049304 
##           315           316           317           318           319 
##  0.1130158270  0.3954531491 -0.1127523731  0.2839504974 -0.0250946533 
##           320           321           322           323           324 
##  0.0699236576 -0.0873765811  0.1822732678 -0.0524892235 -0.1313221403 
##           325           326           327           328           329 
##  0.1896033194 -0.0379981822 -0.4585078784  0.3170923344  0.8274083898 
##           330           331           332           333           334 
##  0.0170256029  0.3331875864 -0.0956242470  0.2653746905 -0.0868932508 
##           335           336           337           338           339 
##  0.4428391809 -0.1879427310 -0.1522270214  0.0088633796 -0.0994680309 
##           340           341           342           343           344 
## -0.0640388311 -0.1319802778  0.0309371281  0.0970880798  0.0526235860 
##           345           346           347           348           349 
##  0.3642435987 -0.0122921656  0.2983885408  0.2174855792  0.3614189279 
##           350           351           352           353           354 
##  0.2396446189  0.3824789125  0.2370636112  0.4664127077  0.3852501660 
##           355           356           357           358           359 
##  0.3912315500 -0.0534632855 -0.7024882832  0.7315535826  0.5664827855 
##           360           361           362           363           364 
##  0.5103028299  0.4049679414  0.4882264174 -0.1039341230 -0.2937518809 
##           365           366           367           368           369 
##  0.3664235209  0.3806250199  0.2264494251  0.2979040749  0.0046701908 
##           370           371           372           373           374 
## -0.0388709765 -0.3638722967 -0.2258985448  0.2799558245 -0.0833916842 
##           375           376           377           378           379 
##  0.1006425781 -0.0324932439 -0.0414415950 -0.2283320322 -1.6211632191 
##           380           381           382           383           384 
##  0.2155942026  0.5984436769  0.5513846964  0.3097721694  0.0141105734 
##           385           386           387           388           389 
## -0.3504824320 -0.0155012726  0.0550026094 -0.1060995104 -0.0362676052 
##           390           391           392           393           394 
## -0.0669212244 -0.1038404990 -0.2445221009 -0.0321315667  0.1935611142 
##           395           396           397           398           399 
## -0.1990305844  0.1288498802 -0.1395165988  0.0810524377 -0.2233803843 
##           400           401           402           403           404 
## -0.1412669936  0.2346628462 -0.0557033871  0.0035785463  0.0205284953 
##           405           406           407           408           409 
##  0.0575236295 -0.1259511953 -0.6486448488  0.1294277638 -0.3260230531 
##           410           411           412           413           414 
## -0.2641930418 -0.1117633802  0.0399188771 -0.0017608465 -0.0033773442 
##           415           416           417           418           419 
## -0.1698381287 -0.0690817436 -0.3251908652 -0.1149418965  0.4609519715 
##           420           421           422           423           424 
## -0.5726712724 -1.0233111481  0.1999347262 -0.4071212924 -0.3016727155 
##           425           426           427           428           429 
##  0.0929600487 -0.3234051885 -0.5504271923  0.5602989348 -0.4994594928 
##           430           431           432           433           434 
## -0.1733323786 -0.2852291210 -0.6526409194  0.3512642487  0.1878077627 
##           435           436           437           438           439 
## -0.1274407388 -0.3664902188  0.0707644576  0.9624969937 -0.2920790764 
##           440           441           442           443           444 
##  0.0287956529 -0.2419904170  0.2549506243  0.1771499983 -0.1969683895 
##           445           446           447           448           449 
##  0.1487716383  0.2628499292 -0.1876762151 -0.3991412032  0.4333106292 
##           450           451           452           453           454 
## -0.6711290534  1.0230188793  0.0956138993  0.1545294982 -0.3716568823 
##           455           456           457           458           459 
##  0.0648148703  0.0006020451  0.1708435560  0.0955116158  0.2311994734 
##           460           461           462           463           464 
##  0.2781582430 -0.1864942575 -0.4888360821  0.4620323943  0.4795995133 
##           465           466           467           468           469 
##  0.2138113196  0.2799563701  0.3758826737 -0.3528637019  0.0608659002 
##           470           471           472           473           474 
## -0.7088603430  0.5129318160  0.0411735522  0.0752466074  0.1795057709 
##           475           476           477           478           479 
## -0.2067345078 -0.3160843719 -0.0290998807  0.4062518168 -0.0794030696 
##           480           481           482           483           484 
##  0.3694185445 -0.0056020483  0.0583290481  0.0679268293  0.0317987549 
##           485           486           487           488           489 
##  0.1776004512  0.0178536066  0.1775098096  0.0390829284  0.0003528169 
##           490 
##  0.1904342536

remove <- list(p =list(TAVG=c(1:11),mean_precipation=c(1:11)))
mod_ardl1311_weather_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                      TAVG,data = full_cases_wastewater_weather_data_meck_train, 
                                    p=11,q=13,
                                    remove = remove)
f_ardl1311_weather_meck <- forecast(mod_ardl1311_weather_meck, x= t(full_cases_wastewater_weather_data_meck_test[,c(8,5,6)]),h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl1311_weather_meck$forecasts)
## [1] 0.2487683
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl1311_weather_meck$forecasts) 
## [1] 0.2215939
checkresiduals(mod_ardl1311_weather_meck)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
##  0.0158737146  0.1191886122  0.0574314550 -0.1094952895 -0.1645733032 
##            19            20            21            22            23 
## -0.0672699904  0.2563075688  0.0363321861 -0.0009455866 -0.0389911733 
##            24            25            26            27            28 
##  0.2758936892  0.0952516351 -0.1883819807 -0.0333570577  0.2158247308 
##            29            30            31            32            33 
## -0.0100698372 -0.0232497585  0.0296169483 -0.1359132937 -0.0323663445 
##            34            35            36            37            38 
## -0.3107204910  0.4466207935  0.0548089602 -0.0465662334 -0.0483215325 
##            39            40            41            42            43 
## -0.1613841887  0.1639665518  0.1871567400 -0.0048842122 -0.1101190278 
##            44            45            46            47            48 
##  0.1458180436 -0.0553780080 -0.1688691061  0.1855564280 -0.2617507911 
##            49            50            51            52            53 
##  0.3338265209 -0.1990200605 -0.2922552262  0.1767991827 -0.0163890012 
##            54            55            56            57            58 
##  0.0439883480  0.0310437118 -0.1292291167  0.3547549395 -0.3257337222 
##            59            60            61            62            63 
##  0.1761836270 -0.9948175753 -0.3343415233  0.1472259342 -0.5357874370 
##            64            65            66            67            68 
##  0.5309236620  0.3761237716  0.0829560907  0.1656709891  0.2486686713 
##            69            70            71            72            73 
##  0.3510848996  0.2367437083 -0.0927649268  0.0078838295 -0.1338419375 
##            74            75            76            77            78 
##  0.2099364952  0.0521075692  0.3847165935  0.3920960163  0.2809831519 
##            79            80            81            82            83 
## -0.0126571048 -0.0169812612 -0.1719966020 -0.2676293799 -0.1789864947 
##            84            85            86            87            88 
## -0.1040711941  0.2690701781 -0.2729825539  0.4056434410  0.0769264328 
##            89            90            91            92            93 
##  0.2738039807 -0.1952520178 -0.0562393218 -0.0708750471 -0.4677945571 
##            94            95            96            97            98 
## -0.0932064711 -0.2012010955  0.1178310695  0.0165016620  0.0686019854 
##            99           100           101           102           103 
## -0.0118492009 -0.0093642692  0.1912197679 -0.0094658407 -0.3916785598 
##           104           105           106           107           108 
##  0.3007933977 -0.5220230463  0.5658464688 -0.0634008294 -0.1226736376 
##           109           110           111           112           113 
##  0.1246573888 -0.3395573899  0.1097060155  0.3243776394 -0.1744775351 
##           114           115           116           117           118 
## -0.0515667868 -0.1764144834 -0.4177612298  0.3796194006 -0.3818854875 
##           119           120           121           122           123 
## -0.0228050064 -0.2062900875  0.0812646453 -0.1101771454  0.0946238868 
##           124           125           126           127           128 
## -0.3563966159  0.0591460039  0.2551893999 -0.2567295322 -0.6183574270 
##           129           130           131           132           133 
## -0.3455809985 -0.7845314846 -0.0900439215 -0.1384478021  0.6821814842 
##           134           135           136           137           138 
##  0.1204405705  0.1726842640 -0.0542289729 -0.1288603035  0.2426178544 
##           139           140           141           142           143 
## -0.4984207458 -0.2193024030  0.1893662071 -0.3320134899 -0.6201061756 
##           144           145           146           147           148 
##  0.5455639461  0.4139741381  0.0800026652 -0.0026186138 -0.1336317498 
##           149           150           151           152           153 
## -0.3408138696  0.8389974225  0.0435887991 -0.6998545630 -0.4130747487 
##           154           155           156           157           158 
##  0.3009301190 -0.1030574348 -0.2422273098 -0.0816572756  0.2522217610 
##           159           160           161           162           163 
##  0.1417016658 -0.3585542275  0.4501881686  0.3542186698  0.2498923126 
##           164           165           166           167           168 
##  0.2004700631 -0.3199266089 -0.3544502962 -0.2904129132  0.3721171546 
##           169           170           171           172           173 
## -0.6516905737 -0.6914395828 -0.6246059968  0.3860183227 -0.1300577859 
##           174           175           176           177           178 
## -0.6210884036  0.2829620951  0.2381189825 -0.1788815880  0.1251047846 
##           179           180           181           182           183 
## -0.1771922377  0.8311654548  0.2573304351  0.1454530017 -0.3849041120 
##           184           185           186           187           188 
##  0.0182911252 -0.3866730643  0.1027032108  0.3080459288  0.4038257782 
##           189           190           191           192           193 
##  0.4211338248  0.2487343775 -0.1416875521  0.2101165099 -0.2318115563 
##           194           195           196           197           198 
## -0.1499914417 -0.1007594881 -0.2155730486  0.7223806072 -0.1243255148 
##           199           200           201           202           203 
##  0.2132001993  0.1330607866  0.1359933895  0.2219120062  0.1331209204 
##           204           205           206           207           208 
##  0.1334922027 -0.0614375276 -0.2055712660  0.0575336152  0.0544301044 
##           209           210           211           212           213 
## -0.1399901670  0.0960978861  0.1220858351 -0.0315250406 -0.1636483722 
##           214           215           216           217           218 
## -0.0797690500  0.1424769260 -0.0807805676 -0.2108874504  0.1136784514 
##           219           220           221           222           223 
## -0.0991636657  0.0005695359  0.1490375414  0.0641720539  0.0516117587 
##           224           225           226           227           228 
## -0.2557262184  0.0158092476 -0.0325518999  0.1357941130  0.0042893021 
##           229           230           231           232           233 
##  0.1634441170  0.1139710669  0.0789295175  0.1597189708  0.1673319984 
##           234           235           236           237           238 
## -0.1942231608 -0.1803889536 -0.1132511347  0.1042327732 -0.1276703621 
##           239           240           241           242           243 
## -0.0979503539  0.1178464226 -0.0932191341 -0.2045959156  0.0854291315 
##           244           245           246           247           248 
## -0.0070849990 -0.1163745736  0.0225616467 -0.8433853874  0.5396427749 
##           249           250           251           252           253 
##  0.1804158145  0.2798251809  0.1172630687  0.1340594909 -0.1530311930 
##           254           255           256           257           258 
##  0.1422036958 -0.1632186165 -0.3425982138 -0.1821317893 -0.0492521459 
##           259           260           261           262           263 
## -0.1508064317 -0.2036391792  0.0593469658 -0.0501316698 -0.1931030232 
##           264           265           266           267           268 
##  0.1290562021 -0.1163581066 -0.0198323521 -0.3181038788  0.1649761890 
##           269           270           271           272           273 
## -0.2394216331  0.4049603545 -0.0308865762  0.1987983846 -0.1464230101 
##           274           275           276           277           278 
## -0.2488212421  0.0436514055 -0.0676581371 -0.4400354084 -0.1134816001 
##           279           280           281           282           283 
##  0.0206295985 -0.2824440598  0.0110098602  0.0457117095 -0.1181096940 
##           284           285           286           287           288 
##  0.0821548624 -0.1294966210  0.2343558286  0.0780350225 -0.0975202390 
##           289           290           291           292           293 
## -0.1163938941 -0.3473691643 -0.1669149157 -1.1985656861 -0.3466319734 
##           294           295           296           297           298 
##  0.6291373098  0.5916967934  0.1083403147  0.3027695994  0.4596312306 
##           299           300           301           302           303 
## -0.1755506319  0.2235341747  0.2145392956 -0.3513029711 -0.1716800675 
##           304           305           306           307           308 
## -0.1044548103 -0.4579931085  0.1506511341 -0.0636453704  0.1981570648 
##           309           310           311           312           313 
##  0.0295527237  0.1194162166 -0.2157289679  0.0242104586 -0.0383068807 
##           314           315           316           317           318 
##  0.1135450482  0.0458665230  0.2688698147 -0.1325394696  0.2173243088 
##           319           320           321           322           323 
## -0.0105760882  0.0962588459 -0.1452890034  0.0895038732 -0.1068657352 
##           324           325           326           327           328 
## -0.2095346309  0.1629866463 -0.0609950801 -0.4991670667  0.2410300166 
##           329           330           331           332           333 
##  0.8105635466  0.0531705353  0.3747566143 -0.0282769714  0.3151841861 
##           334           335           336           337           338 
## -0.0545672954  0.4762975470 -0.1873769151 -0.2565359099 -0.0630189946 
##           339           340           341           342           343 
## -0.0827415812 -0.0385979326 -0.1277227849  0.0852038521  0.0506385085 
##           344           345           346           347           348 
## -0.0128577471  0.3944445607  0.0472551578  0.2805069026  0.1921452034 
##           349           350           351           352           353 
##  0.3830751235  0.2043808145  0.3132038192  0.1898493332  0.3688528703 
##           354           355           356           357           358 
##  0.3070221289  0.3561671038 -0.1162356295 -0.8077344548  0.6183725338 
##           359           360           361           362           363 
##  0.4839856828  0.5023002081  0.3650093659  0.4086356492 -0.1349365882 
##           364           365           366           367           368 
## -0.3058305809  0.3591279580  0.2987514275  0.0657781302  0.2052432923 
##           369           370           371           372           373 
##  0.0870807256  0.0507408014 -0.2370787470 -0.1304178007  0.2981116418 
##           374           375           376           377           378 
## -0.1591948174  0.0348666159  0.0018689451  0.1015877929 -0.1683735809 
##           379           380           381           382           383 
## -1.6681054055  0.2444465826  0.4945394142  0.4686368846  0.4236487965 
##           384           385           386           387           388 
##  0.0866157284 -0.2472811680  0.1519131274  0.2506312144 -0.0530560030 
##           389           390           391           392           393 
## -0.2240944661 -0.2658850637 -0.1194835269 -0.2118159491  0.0619235780 
##           394           395           396           397           398 
##  0.2358100783 -0.2578144670  0.0963706791 -0.0779806537  0.1016122469 
##           399           400           401           402           403 
## -0.2044577917 -0.1074287878  0.2132434815 -0.1168794108 -0.0091786027 
##           404           405           406           407           408 
##  0.0545625358  0.0425910357 -0.0958359137 -0.5192578616  0.1672377594 
##           409           410           411           412           413 
## -0.3018072864 -0.2363497605 -0.0416518207  0.0427226727  0.0624456278 
##           414           415           416           417           418 
##  0.1846123641 -0.0280293762  0.0601456006 -0.2910083783 -0.1418309132 
##           419           420           421           422           423 
##  0.4589889351 -0.5769359760 -0.9588011142  0.2068689738 -0.3477767305 
##           424           425           426           427           428 
## -0.2012181232  0.1988488575 -0.2922556261 -0.4820601576  0.7467401459 
##           429           430           431           432           433 
## -0.2651484874  0.0268347445 -0.2684883340 -0.6899292206  0.3422417228 
##           434           435           436           437           438 
##  0.1582684440 -0.0375826477 -0.3841446834 -0.0181625459  0.9886537225 
##           439           440           441           442           443 
## -0.1638454244  0.0950672555 -0.2371768380  0.1403236707  0.1225477020 
##           444           445           446           447           448 
## -0.1760909432  0.1190934209  0.1336165703 -0.3424671647 -0.4271470724 
##           449           450           451           452           453 
##  0.4677305362 -0.7400281423  1.0217398401  0.0783359554  0.0816893328 
##           454           455           456           457           458 
## -0.4090480932  0.0042204539  0.0140112817  0.0854735645 -0.0323518619 
##           459           460           461           462           463 
##  0.1022436393  0.1984354308 -0.1993437734 -0.4165408393  0.3997853001 
##           464           465           466           467           468 
##  0.4143609060  0.1289833992  0.2344193175  0.4038177123 -0.3690978430 
##           469           470           471           472           473 
##  0.0212369229 -0.6549080304  0.4145359880 -0.1230323563 -0.0634018220 
##           474           475           476           477           478 
##  0.1551319643 -0.2349890515 -0.3120571059 -0.0006542899  0.4892901762 
##           479           480           481           482           483 
## -0.1339658176  0.1679545588 -0.1141280845  0.0117768968  0.0305786997 
##           484           485           486           487           488 
##  0.0545905569  0.1469298736 -0.1144001991  0.0931871094  0.0550377171 
##           489           490 
##  0.0064183479  0.1455704804

remove <- list(p =list(TAVG=c(1:14),mean_precipation=c(1:14)))
mod_ardl914_weather_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                       TAVG,data = full_cases_wastewater_weather_data_meck_train, 
                                     p=14,q=9,
                                     remove = remove)

remove <- list(p =list(TAVG=c(1:14),mean_precipation=c(1:14)))
mod_ardl914_weather_meck <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                       TAVG,data = full_cases_wastewater_weather_data_meck_train, 
                                     p=14,q=9,
                                     remove = remove)
summary(mod_ardl914_weather_meck)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.60579 -0.16172  0.00929  0.17423  1.01182 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.340915   0.488841  -0.697 0.485916    
## log_viral_gene.t     -0.062055   0.035360  -1.755 0.079954 .  
## log_viral_gene.1      0.119283   0.046192   2.582 0.010129 *  
## log_viral_gene.2     -0.048323   0.046095  -1.048 0.295051    
## log_viral_gene.3      0.115203   0.046173   2.495 0.012953 *  
## log_viral_gene.4     -0.109250   0.046488  -2.350 0.019202 *  
## log_viral_gene.5      0.068536   0.046770   1.465 0.143513    
## log_viral_gene.6     -0.071897   0.047002  -1.530 0.126800    
## log_viral_gene.7      0.048087   0.046624   1.031 0.302918    
## log_viral_gene.8     -0.013162   0.046546  -0.283 0.777472    
## log_viral_gene.9     -0.007683   0.046473  -0.165 0.868757    
## log_viral_gene.10     0.015626   0.045933   0.340 0.733877    
## log_viral_gene.11     0.034814   0.045823   0.760 0.447800    
## log_viral_gene.12    -0.048558   0.045715  -1.062 0.288720    
## log_viral_gene.13    -0.072942   0.045146  -1.616 0.106861    
## log_viral_gene.14     0.049557   0.034603   1.432 0.152792    
## mean_precipation.t   -0.086905   0.059140  -1.469 0.142404    
## TAVG.t                0.001617   0.001151   1.406 0.160507    
## log_mean_new_cases.1  0.515709   0.047062  10.958  < 2e-16 ***
## log_mean_new_cases.2  0.050023   0.052925   0.945 0.345083    
## log_mean_new_cases.3  0.207197   0.052849   3.921 0.000102 ***
## log_mean_new_cases.4  0.055063   0.053651   1.026 0.305300    
## log_mean_new_cases.5  0.136893   0.053930   2.538 0.011476 *  
## log_mean_new_cases.6  0.076001   0.053685   1.416 0.157556    
## log_mean_new_cases.7  0.002560   0.052938   0.048 0.961449    
## log_mean_new_cases.8  0.011205   0.053208   0.211 0.833299    
## log_mean_new_cases.9 -0.089535   0.047957  -1.867 0.062554 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3118 on 449 degrees of freedom
## Multiple R-squared:  0.9248, Adjusted R-squared:  0.9205 
## F-statistic: 212.4 on 26 and 449 DF,  p-value: < 2.2e-16
f_ardl914_weather_meck <- forecast(mod_ardl914_weather_meck, 
                                   x= t(full_cases_wastewater_weather_data_meck_test[,c(8,5,6)]),
                                   h=14,
                                   interval = TRUE)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_ardl914_weather_meck$forecasts[,2]) 
## [1] 0.1278721
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_ardl914_weather_meck$forecasts[,2]) 
## [1] 0.1061923
checkresiduals(mod_ardl914_weather_meck)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
##  0.0201283190  0.0488150294 -0.1212445118 -0.1326995689 -0.0260273329 
##            20            21            22            23            24 
##  0.2363100520 -0.0069240528  0.0325582815 -0.0439697482  0.2647168202 
##            25            26            27            28            29 
##  0.0822631479 -0.2285264329 -0.0432912887  0.2847376330  0.0670551627 
##            30            31            32            33            34 
## -0.0408452932  0.0110197700 -0.1463728842 -0.0486978196 -0.3357647354 
##            35            36            37            38            39 
##  0.4415179971  0.0391517393 -0.0725094928 -0.1077128360 -0.2280820437 
##            40            41            42            43            44 
##  0.1799130977  0.1427038608 -0.1175441095 -0.1253020780  0.2619775775 
##            45            46            47            48            49 
## -0.0296655180 -0.1984936111  0.1660925722 -0.2657746868  0.4203046113 
##            50            51            52            53            54 
## -0.1704374764 -0.3356754813  0.1579798005 -0.0221529676  0.0243912911 
##            55            56            57            58            59 
##  0.0264425661 -0.1051552698  0.3299146517 -0.3036299290  0.1765093991 
##            60            61            62            63            64 
## -1.0413487663 -0.3008866089  0.2000447140 -0.4356764335  0.6144193380 
##            65            66            67            68            69 
##  0.3455960693  0.1339947187  0.1338025037  0.2564116715  0.2944261365 
##            70            71            72            73            74 
##  0.1859188551  0.0044690629  0.0291801799 -0.0757405338  0.2407017225 
##            75            76            77            78            79 
##  0.0621641989  0.3590910071  0.2465377375  0.1916996835 -0.1167738608 
##            80            81            82            83            84 
## -0.1170935287 -0.2590382822 -0.4345162313 -0.1564978651  0.1124205770 
##            85            86            87            88            89 
##  0.3529894322 -0.2180826280  0.3850756895  0.0235386862  0.2154448141 
##            90            91            92            93            94 
## -0.2333332317 -0.0758013935 -0.0092868825 -0.3902992019 -0.0149761140 
##            95            96            97            98            99 
## -0.1888239975  0.1272603703  0.0108878584  0.0605564862 -0.0498895291 
##           100           101           102           103           104 
## -0.0653001936  0.1725358448 -0.0720018916 -0.3440465910  0.3256178382 
##           105           106           107           108           109 
## -0.6132325043  0.5932079887 -0.0307810632 -0.1299989369  0.1330739435 
##           110           111           112           113           114 
## -0.2921760430  0.1217264632  0.2577252518 -0.1108776203 -0.0568978445 
##           115           116           117           118           119 
## -0.1500607459 -0.4220268469  0.2946226254 -0.4004986241 -0.0773630385 
##           120           121           122           123           124 
## -0.2053811059  0.1080115271 -0.1848536325  0.0194957682 -0.3977712711 
##           125           126           127           128           129 
## -0.0108496389  0.2079484992 -0.2651016708 -0.5713934918 -0.2665189185 
##           130           131           132           133           134 
## -0.6987143002  0.0220496472 -0.0819594102  0.6430981555  0.1291788495 
##           135           136           137           138           139 
##  0.1555141041 -0.2256247425 -0.3182154388  0.2458704973 -0.5753080221 
##           140           141           142           143           144 
## -0.2447128990  0.2864609232 -0.3102346320 -0.6472301657  0.4545869389 
##           145           146           147           148           149 
##  0.2888571398  0.0046638524  0.0677896022 -0.1436797761 -0.2975435328 
##           150           151           152           153           154 
##  0.9302083551 -0.0478129245 -0.6818959202 -0.3007382440  0.2227184605 
##           155           156           157           158           159 
## -0.1616353022 -0.1546323230 -0.1055656302  0.1200162040  0.1826333643 
##           160           161           162           163           164 
## -0.4303634296  0.2756662181  0.2110187825  0.3555622651  0.3142907940 
##           165           166           167           168           169 
## -0.4250075334 -0.3458685426 -0.2156838276  0.4979862007 -0.6254678315 
##           170           171           172           173           174 
## -0.5871599468 -0.4951690596  0.4226674172 -0.1374607712 -0.7125665982 
##           175           176           177           178           179 
##  0.2108929774  0.2564666159 -0.2388332648 -0.0424545135 -0.2487124340 
##           180           181           182           183           184 
##  0.9244455776  0.2521773160  0.0320575630 -0.3748313302  0.1091830472 
##           185           186           187           188           189 
## -0.2448141070  0.1253664267  0.3404142609  0.4132420921  0.5462058477 
##           190           191           192           193           194 
##  0.3309711639 -0.2234567977  0.1895366953 -0.1882241425 -0.1086226271 
##           195           196           197           198           199 
## -0.0002795538 -0.0568951368  0.8427797233 -0.1044659888  0.1457173324 
##           200           201           202           203           204 
##  0.0517476540  0.1855636861  0.2121105867  0.1422296510  0.1961586697 
##           205           206           207           208           209 
## -0.0197928331 -0.1472176952  0.0726175850  0.1154343015 -0.0689374469 
##           210           211           212           213           214 
##  0.1357691734  0.1499241883 -0.0314317300 -0.2329401572 -0.1256030631 
##           215           216           217           218           219 
##  0.1633746729 -0.0374837702 -0.1833350212  0.0903962117 -0.0711035508 
##           220           221           222           223           224 
## -0.0228528772  0.1344623395  0.0810455426  0.0289270733 -0.2191975792 
##           225           226           227           228           229 
##  0.0417125843 -0.0390216287  0.1895948591  0.0002243875  0.1349596695 
##           230           231           232           233           234 
##  0.1563756467  0.0641035810  0.0866483768  0.1034920253 -0.2541776775 
##           235           236           237           238           239 
## -0.1843944828 -0.0723875999  0.1450551646 -0.1128776869 -0.1157265536 
##           240           241           242           243           244 
##  0.0942295273 -0.0995244525 -0.2171402609  0.0047886619 -0.0868715720 
##           245           246           247           248           249 
## -0.1291132014  0.0437669475 -0.8928762829  0.5112494993  0.1518523876 
##           250           251           252           253           254 
##  0.2029801182  0.0259306496  0.1178250176 -0.1620502577  0.0958860781 
##           255           256           257           258           259 
## -0.1891556641 -0.3873468839 -0.0434896657  0.0614287367 -0.1435505630 
##           260           261           262           263           264 
## -0.1943626312  0.0464628027 -0.0676627395 -0.1836972247  0.0978056988 
##           265           266           267           268           269 
## -0.1619764374 -0.0066188676 -0.3003855493  0.1427711298 -0.2749812721 
##           270           271           272           273           274 
##  0.4054028500 -0.0585238430  0.1720090134 -0.1435544863 -0.3111503525 
##           275           276           277           278           279 
##  0.0151947242 -0.0784707326 -0.4203035576 -0.1209718526  0.0853501505 
##           280           281           282           283           284 
## -0.2630570451 -0.0097176033  0.0217739055 -0.1805067177  0.0894963125 
##           285           286           287           288           289 
## -0.1290611624  0.1878877812  0.0883200902 -0.0987111860 -0.1539851700 
##           290           291           292           293           294 
## -0.3058282129 -0.1394998610 -1.2657394464 -0.2964238430  0.6556146678 
##           295           296           297           298           299 
##  0.5640545389  0.0397205897  0.1567183869  0.3413238396 -0.2769593710 
##           300           301           302           303           304 
##  0.1839986251  0.1582594591 -0.2607158111  0.0688198927  0.0030143123 
##           305           306           307           308           309 
## -0.4678111034  0.1604046778 -0.0216893350  0.1851310486  0.0643104666 
##           310           311           312           313           314 
##  0.1669077490 -0.2360229824  0.0301671399 -0.0678648256  0.0184729591 
##           315           316           317           318           319 
##  0.1150075333  0.3836581991 -0.1093838493  0.2765331599 -0.0147658842 
##           320           321           322           323           324 
##  0.0424690048 -0.1035010061  0.1593947591 -0.0594280641 -0.1482651978 
##           325           326           327           328           329 
##  0.2054185576 -0.0611675068 -0.4521366763  0.3088948699  0.8317678125 
##           330           331           332           333           334 
##  0.0094925732  0.3444435840 -0.0803254738  0.2535385979 -0.0751852975 
##           335           336           337           338           339 
##  0.4518706579 -0.2383406292 -0.1917932280  0.0511779478 -0.1080730236 
##           340           341           342           343           344 
## -0.0486817126 -0.1496596507  0.0432020769  0.0758517456  0.0848001098 
##           345           346           347           348           349 
##  0.3707296165 -0.0330324572  0.3021538905  0.2079225313  0.3525140059 
##           350           351           352           353           354 
##  0.2311489351  0.3682833555  0.2249327910  0.4246291085  0.3790118669 
##           355           356           357           358           359 
##  0.3584425530 -0.0723131399 -0.7280857993  0.7183999157  0.5446988555 
##           360           361           362           363           364 
##  0.5054746228  0.3910730824  0.4835643544 -0.0978374034 -0.2828114955 
##           365           366           367           368           369 
##  0.3764994366  0.2677914730  0.2207114119  0.3164008377  0.0274361339 
##           370           371           372           373           374 
## -0.0072345778 -0.3303328434 -0.2148824675  0.2411740335 -0.1052125927 
##           375           376           377           378           379 
##  0.1180519765 -0.0208608354 -0.0135154079 -0.2076882045 -1.6057928545 
##           380           381           382           383           384 
##  0.2023905050  0.5626318024  0.5843278780  0.3296332474  0.0325247466 
##           385           386           387           388           389 
## -0.3158350731  0.0138877175  0.1206834505 -0.2325783333 -0.0899623800 
##           390           391           392           393           394 
## -0.0515088696 -0.0800730623 -0.2081954678 -0.0149166863  0.1628597724 
##           395           396           397           398           399 
## -0.2144053530  0.1487403432 -0.1266828330  0.0891185964 -0.2150317425 
##           400           401           402           403           404 
## -0.1405795408  0.2247485400 -0.0714916600  0.0310274271  0.0095112489 
##           405           406           407           408           409 
##  0.0721484129 -0.1139767239 -0.6319492888  0.1347349159 -0.3339801720 
##           410           411           412           413           414 
## -0.2350850446 -0.0925018697  0.0519996810  0.0262323211  0.0239706564 
##           415           416           417           418           419 
## -0.1210319847 -0.0902867793 -0.2969519786 -0.1089753036  0.4488886232 
##           420           421           422           423           424 
## -0.5614778179 -1.0145384308  0.2060284186 -0.3726424204 -0.2782998785 
##           425           426           427           428           429 
##  0.1281983169 -0.3079515360 -0.5520580365  0.6365216116 -0.4493224737 
##           430           431           432           433           434 
## -0.2112375409 -0.2488498463 -0.6439388068  0.3456390686  0.2083050044 
##           435           436           437           438           439 
## -0.1158269722 -0.4284632083  0.1220175448  0.9734936680 -0.2989246798 
##           440           441           442           443           444 
##  0.0438419779 -0.2955115509  0.2414808957  0.1959026982 -0.2008684833 
##           445           446           447           448           449 
##  0.1100371765  0.1986834017 -0.1385911511 -0.4222284458  0.4183589866 
##           450           451           452           453           454 
## -0.7059813316  1.0118186719  0.1168122849  0.1405620212 -0.3802367067 
##           455           456           457           458           459 
##  0.0667790796 -0.0077305667  0.0975681065  0.1293569262  0.1358811582 
##           460           461           462           463           464 
##  0.3141217011 -0.1658673783 -0.4843261887  0.4207965053  0.4549704123 
##           465           466           467           468           469 
##  0.1952991999  0.2796675198  0.3572061322 -0.3522744006  0.0790238488 
##           470           471           472           473           474 
## -0.7132441820  0.4320866851  0.0090954243  0.0841965604  0.1804996445 
##           475           476           477           478           479 
## -0.2165604539 -0.2766053560 -0.0626253799  0.4199812622 -0.1489028740 
##           480           481           482           483           484 
##  0.3668738278 -0.0042531929  0.0368610670  0.0795556153  0.0152495691 
##           485           486           487           488           489 
##  0.1352920769 -0.0227313652  0.2013421224  0.0243414321  0.0052767780 
##           490 
##  0.1734647060

exp(f_ardl914_weather_meck$forecasts[1,2])
## [1] 3.629081
exp(f_ardl914_weather_meck$forecasts[1,1])
## [1] 2.013232
exp(f_ardl914_weather_meck$forecasts[1,3])
## [1] 6.643546
exp(f_ardl914_weather_meck$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_meck_test[1,7])
## [1] 0.6029537
exp(f_ardl914_weather_meck$forecasts[7,2])
## [1] 3.720462
exp(f_ardl914_weather_meck$forecasts[7,1])
## [1] 1.614414
exp(f_ardl914_weather_meck$forecasts[7,3])
## [1] 9.171205
exp(f_ardl914_weather_meck$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_meck_test[7,7])
## [1] -0.1339966
exp(f_ardl914_weather_meck$forecasts[14,2])
## [1] 3.909254
exp(f_ardl914_weather_meck$forecasts[14,1])
## [1] 1.261592
exp(f_ardl914_weather_meck$forecasts[14,3])
## [1] 11.72392
exp(f_ardl914_weather_meck$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_meck_test[14,7])
## [1] 0.8893137
#New Hanover

full_cases_wastewater_weather_data_hanover <- 
  full_cases_wastewater_weather_data_hanover[-c(505,506,507),]

full_cases_wastewater_weather_data_hanover <- full_cases_wastewater_weather_data_hanover %>% 
  mutate(log_mean_new_cases = log(mean_new_cases),
         log_viral_gene = log(full_viral_gene_copies_per_person))

full_cases_wastewater_weather_data_hanover <- full_cases_wastewater_weather_data_hanover %>% 
  mutate(log_mean_new_cases = seasadj(decompose(ts(log_mean_new_cases, frequency=7))),
         log_viral_gene = seasadj(decompose(ts(log_viral_gene, frequency=7))))

full_cases_wastewater_weather_data_hanover_train <- 
  full_cases_wastewater_weather_data_hanover[-c(491:504),]
full_cases_wastewater_weather_data_hanover_test <- 
  full_cases_wastewater_weather_data_hanover[c(491:504),]

lowest_rmse_hanover <- Inf
best_mod_hanover <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                   data = full_cases_wastewater_weather_data_hanover_train, p=p,q=q)
    f <- forecast(mod, x= t(full_cases_wastewater_weather_data_hanover_test[,7]),h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
                         f$forecasts) #interchanged between RMSE and MAE 
    if (forecast_acc<lowest_rmse_hanover){
      lowest_rmse_hanover<- forecast_acc
      best_mod_hanover <-mod 
    }
  }
}

lowest_rmse_hanover #0.348 (0.317)
## [1] 0.3413698
summary(best_mod_hanover) 
## 
## Time series regression with "ts" data:
## Start = 10, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.48883 -0.21033  0.00741  0.23862  1.15177 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.683459   0.225717  -3.028  0.00260 ** 
## log_viral_gene.t      0.073524   0.029404   2.500  0.01274 *  
## log_viral_gene.1     -0.023619   0.040463  -0.584  0.55969    
## log_viral_gene.2     -0.001439   0.029774  -0.048  0.96148    
## log_mean_new_cases.1  0.457334   0.045959   9.951  < 2e-16 ***
## log_mean_new_cases.2  0.123586   0.050710   2.437  0.01518 *  
## log_mean_new_cases.3  0.046925   0.050427   0.931  0.35256    
## log_mean_new_cases.4  0.150204   0.050320   2.985  0.00298 ** 
## log_mean_new_cases.5  0.070798   0.050737   1.395  0.16356    
## log_mean_new_cases.6  0.074289   0.050361   1.475  0.14085    
## log_mean_new_cases.7  0.155353   0.050377   3.084  0.00216 ** 
## log_mean_new_cases.8 -0.058520   0.050537  -1.158  0.24746    
## log_mean_new_cases.9 -0.100652   0.045641  -2.205  0.02792 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3794 on 468 degrees of freedom
## Multiple R-squared:  0.9009, Adjusted R-squared:  0.8984 
## F-statistic: 354.7 on 12 and 468 DF,  p-value: < 2.2e-16
tsdisplay(residuals(best_mod_hanover))
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -0.2937247715  0.0725525051  0.2357132291  0.0428823729  0.2024050180 
##            15            16            17            18            19 
##  0.1383610625 -0.2676940703 -0.0134268985 -0.0117135696  0.0764144973 
##            20            21            22            23            24 
##  0.1981303419  0.2113455282  0.2379975457  0.0051074184 -0.0491099861 
##            25            26            27            28            29 
## -0.1261627848 -0.2905892448  0.6591943825 -0.2276926085  0.3194060555 
##            30            31            32            33            34 
## -0.4551185306  0.1445597583 -0.0568366994 -0.0627825150  0.2938045965 
##            35            36            37            38            39 
##  0.3709020208  0.2960118006 -0.0942220382 -0.1734517743  0.0226241743 
##            40            41            42            43            44 
##  0.0270041027  0.1869636936 -0.2392370427 -0.3142958568  0.2411378355 
##            45            46            47            48            49 
## -0.0771738354 -0.1404487333 -0.3677277543  0.0431663383 -0.4316978812 
##            50            51            52            53            54 
##  0.2531292334 -0.1388129807 -0.4180790801  0.3174781199 -0.1916208740 
##            55            56            57            58            59 
##  0.1987007527  0.3663651337  0.2555273155  0.0239482523 -0.0530943676 
##            60            61            62            63            64 
##  0.1176959277  0.0485663629  0.1937034479  0.0685978034 -0.1774245281 
##            65            66            67            68            69 
## -0.4345144209  0.6653127842  0.1917715091  0.1800176281 -0.2190088185 
##            70            71            72            73            74 
##  0.4411357528  0.0434507919 -0.0425528122  0.0906556734 -0.1789678336 
##            75            76            77            78            79 
##  0.1817007557  0.4857756943  0.2444219798  0.1366642375  0.1156306111 
##            80            81            82            83            84 
##  0.4289515301  0.3184282752  0.1221898253 -0.0488892873 -0.1376011723 
##            85            86            87            88            89 
##  0.2397033861 -0.0037757774  0.2027015932 -0.3899420104 -0.0166538988 
##            90            91            92            93            94 
## -0.2101998536  0.0999245294 -0.2764622533  0.1196021546 -0.1969040255 
##            95            96            97            98            99 
##  0.4246182427  0.0866703371 -0.2692966292  0.1022241170  0.1845049143 
##           100           101           102           103           104 
##  0.2844814622  0.0567238754  0.0718908996 -0.0193711583  0.3085424980 
##           105           106           107           108           109 
## -0.2717931726 -1.1953301375  0.6730721267 -0.1633869109  0.3155728050 
##           110           111           112           113           114 
## -0.0091031336  0.3984023024  0.5548959808 -1.0566851701  0.7880259146 
##           115           116           117           118           119 
##  0.0939056670  0.1223389249  0.0491446447  0.1406920939 -0.6743811810 
##           120           121           122           123           124 
##  0.3197780047  0.2157982532 -0.5256451527  0.0489862102 -1.0499510287 
##           125           126           127           128           129 
##  0.4394807412  0.3652649330 -0.5195959156  0.3998504377 -0.1191334670 
##           130           131           132           133           134 
##  0.4433747581 -0.0947424623  0.0832120425 -0.5277903645 -0.1063871747 
##           135           136           137           138           139 
##  0.2841217598 -0.8616391162 -0.3790968311  0.7288172894  0.5071672677 
##           140           141           142           143           144 
## -0.2286002821  0.0314926421 -0.6829332184  0.9703729179 -0.7449079747 
##           145           146           147           148           149 
##  0.2679769867 -0.3756868031  0.2328898204  0.2944174911 -0.4603615280 
##           150           151           152           153           154 
## -0.2255009541 -0.0717132127 -0.1898146588  0.0234792637  0.3163682613 
##           155           156           157           158           159 
##  0.2111669824 -0.2559877586 -0.0066921052 -0.1121490904 -0.0041986795 
##           160           161           162           163           164 
##  0.0424280537  0.3866164312  0.3249911181 -0.4301015712 -0.0217016916 
##           165           166           167           168           169 
## -0.2131773712  0.0505797912  0.0663116210  0.2714930485  0.2043053345 
##           170           171           172           173           174 
## -0.3305856683  0.0199390521 -0.0987236863 -0.0879896828 -0.1770255319 
##           175           176           177           178           179 
##  0.9075754923 -0.1057388530 -0.4752423733 -0.1321610850 -0.3801484525 
##           180           181           182           183           184 
##  0.5419094734 -0.2656456412  0.1227014611  0.2842934188 -0.3350889685 
##           185           186           187           188           189 
##  0.5033548382 -0.0068548374 -0.6091154625 -0.2503721300  0.0775616818 
##           190           191           192           193           194 
##  0.8245964949 -0.3712032240 -0.6583240983  1.1415074891  0.0754030017 
##           195           196           197           198           199 
##  0.6061736284 -0.2877254417 -0.7529277730 -0.8836332442  0.7384886742 
##           200           201           202           203           204 
##  0.0897824100  0.7422410308  0.3995787787  0.7494566948  0.8276606603 
##           205           206           207           208           209 
##  0.1200761219 -0.0654303538 -0.2230863545  0.0274122666 -0.0400149964 
##           210           211           212           213           214 
##  0.0022512118  0.2936783139  0.0990719213  0.0543947860 -0.0493149906 
##           215           216           217           218           219 
## -0.0373004111  0.3833824395  0.0185383025  0.0589516846  0.0781896085 
##           220           221           222           223           224 
##  0.1379129373 -0.0118587807 -0.1957035215  0.1029201034 -0.1096721278 
##           225           226           227           228           229 
##  0.2728494624  0.2057127592 -0.1336138055  0.0636875549  0.2787792813 
##           230           231           232           233           234 
## -0.0819055576 -0.2028231242  0.1279166587  0.6104019865  0.5059773785 
##           235           236           237           238           239 
##  0.2927907643  0.3309247297 -0.2103299644 -0.6020374951 -0.2870039810 
##           240           241           242           243           244 
## -0.1242200552  0.1387655073  0.1208341292  0.1154879578 -0.1883789764 
##           245           246           247           248           249 
## -0.1700820686 -0.4658964526 -0.8623299961  0.6775253499  0.0105449122 
##           250           251           252           253           254 
## -0.0087829510  0.1708893966 -0.2762265336 -0.0240055186  0.3459538609 
##           255           256           257           258           259 
## -0.1418843551 -0.1564006393 -0.1053706580  0.1066407105 -0.1853499335 
##           260           261           262           263           264 
## -0.2487433641 -0.2333642650 -0.2727303989 -0.1687733447 -0.0466078025 
##           265           266           267           268           269 
## -0.3995324342 -0.0677488924 -0.0471187115 -0.0601087398 -0.1041610652 
##           270           271           272           273           274 
## -0.2367197351 -0.3989378404  0.2484457684 -0.1568777389 -0.7453024532 
##           275           276           277           278           279 
##  0.3555214858  0.1002671666 -0.3868654278  0.3040895309 -0.2028461658 
##           280           281           282           283           284 
##  0.1722189512 -0.2332142707 -0.0440658053 -0.3086458206 -0.0504900649 
##           285           286           287           288           289 
##  0.1012008512  0.1700433893 -0.3481999153 -0.3257685419  0.0074100728 
##           290           291           292           293           294 
##  0.0341751172 -1.4888343080 -0.7335710249 -0.5480315814  0.7716219916 
##           295           296           297           298           299 
## -0.3682446885 -0.0486605721 -0.1022491050  0.4153375971 -0.6283894824 
##           300           301           302           303           304 
##  0.2012134868 -0.5205160486 -0.1192180476 -0.0939335491  0.4307161292 
##           305           306           307           308           309 
## -0.1346345308 -0.6418514036 -0.1222579844  0.1329938855  0.1042333559 
##           310           311           312           313           314 
##  0.5206664141 -0.6846115233 -0.3201812611  0.0842644910  0.6126850050 
##           315           316           317           318           319 
##  0.5875763431 -0.4078496593  0.2353657459  0.3752687472  0.3007912347 
##           320           321           322           323           324 
##  0.0219233387 -0.5585097529  0.0920338782  0.4985910974 -0.2534512110 
##           325           326           327           328           329 
## -0.1352546575 -0.0338617618 -1.2401356127 -0.0603990799  0.6927204590 
##           330           331           332           333           334 
##  0.8101645216  0.4554625066  0.2758977507 -0.1558902425  0.2324918101 
##           335           336           337           338           339 
## -0.1571132546 -0.3554635846  0.1720243647 -0.0368614666 -0.2860883750 
##           340           341           342           343           344 
##  0.3664146703  0.3245726439 -0.8113220120  0.3094211668  0.1261280314 
##           345           346           347           348           349 
##  0.1013780624  0.5355813115 -0.2189059879  0.4203520863  0.4150589674 
##           350           351           352           353           354 
##  0.3031770151  0.1136186294  0.3525790847  0.0603945804  0.6853335007 
##           355           356           357           358           359 
##  0.3605603211 -0.5404365110 -0.6752436627  0.5970516429  0.8297212770 
##           360           361           362           363           364 
##  0.7278450588  0.5652834562  0.5630755331 -0.1175717912 -0.1903461945 
##           365           366           367           368           369 
##  0.0972404074  0.3422963065  0.3265283580  0.0821512567  0.4221394520 
##           370           371           372           373           374 
##  0.3144995798  0.0242426290 -0.0869549654  0.1292901847  0.0371106374 
##           375           376           377           378           379 
##  0.1510399797  0.1887376894  0.2498141128 -0.1476159290 -0.2584843004 
##           380           381           382           383           384 
##  0.2742931587  0.3827214873  0.2861100347  0.0320705553 -0.8687612922 
##           385           386           387           388           389 
## -0.6717371112  0.0901594979  0.3829023442 -0.0934348061 -0.0938720504 
##           390           391           392           393           394 
##  0.1243421596  0.0823670432 -0.4012690476 -0.3010281636 -0.0453070090 
##           395           396           397           398           399 
## -0.1682664417 -0.0227266962 -0.1872004916 -0.1452224339 -0.5144250145 
##           400           401           402           403           404 
## -0.1922387100 -0.0878008475 -0.0624489138 -0.2762842108  0.0024621932 
##           405           406           407           408           409 
## -0.0521510737 -0.7394339480  0.0295910486 -0.2974149703 -0.4524329524 
##           410           411           412           413           414 
## -0.0942575808 -0.2679268279 -0.1104944518 -0.5897524445  0.3438203257 
##           415           416           417           418           419 
## -0.1288860696 -0.2150139160 -0.0303903430  0.0037787687 -0.4397751792 
##           420           421           422           423           424 
## -0.6832706400 -0.4453951363 -0.1344499452 -0.8179229615 -0.4643839583 
##           425           426           427           428           429 
##  0.5094620775 -0.3647641012  0.3028159342  0.2062176388  0.1865326352 
##           430           431           432           433           434 
## -0.5719646627  0.1892242928  0.0735899417 -0.4111611070  0.2540204491 
##           435           436           437           438           439 
##  0.0494242221 -0.5904800236 -0.1388350201 -0.1684062474 -0.1357092483 
##           440           441           442           443           444 
##  0.1638572878  0.4641277285  0.9046284552 -0.5630378051 -0.2394969869 
##           445           446           447           448           449 
## -0.1976548938 -0.2692456496  0.0486816515  0.2693518653  0.1750217782 
##           450           451           452           453           454 
## -0.3212230719 -0.1564946816 -0.2735179742 -0.1829370659 -0.0562354176 
##           455           456           457           458           459 
##  0.2751055137  0.1558797044  0.1341932450 -0.3953978283  0.2066401363 
##           460           461           462           463           464 
##  0.6027321415 -0.6998549402  0.0009059039 -0.0013204327  0.0471160495 
##           465           466           467           468           469 
##  0.3756997387  0.7391826271 -0.3535968665 -0.1872812377 -0.4631693409 
##           470           471           472           473           474 
## -0.4225914438  0.2121285303  0.0406720251 -0.3903567773 -0.2813712050 
##           475           476           477           478           479 
## -0.5980437445  1.1517746897 -0.7341497203  0.2773720907  0.3208178831 
##           480           481           482           483           484 
##  0.3306062548  0.3199183622  0.5542635356 -0.8634054777 -0.1105232295 
##           485           486           487           488           489 
##  0.2386241573 -0.0339323640  0.1725353984 -0.1361991729  0.3281013347 
##           490 
## -0.0849704518

mod_ardl92_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
               data = full_cases_wastewater_weather_data_hanover_train, 
               p=2,q=9)
summary(mod_ardl92_hanover)
## 
## Time series regression with "ts" data:
## Start = 10, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.48883 -0.21033  0.00741  0.23862  1.15177 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.683459   0.225717  -3.028  0.00260 ** 
## log_viral_gene.t      0.073524   0.029404   2.500  0.01274 *  
## log_viral_gene.1     -0.023619   0.040463  -0.584  0.55969    
## log_viral_gene.2     -0.001439   0.029774  -0.048  0.96148    
## log_mean_new_cases.1  0.457334   0.045959   9.951  < 2e-16 ***
## log_mean_new_cases.2  0.123586   0.050710   2.437  0.01518 *  
## log_mean_new_cases.3  0.046925   0.050427   0.931  0.35256    
## log_mean_new_cases.4  0.150204   0.050320   2.985  0.00298 ** 
## log_mean_new_cases.5  0.070798   0.050737   1.395  0.16356    
## log_mean_new_cases.6  0.074289   0.050361   1.475  0.14085    
## log_mean_new_cases.7  0.155353   0.050377   3.084  0.00216 ** 
## log_mean_new_cases.8 -0.058520   0.050537  -1.158  0.24746    
## log_mean_new_cases.9 -0.100652   0.045641  -2.205  0.02792 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3794 on 468 degrees of freedom
## Multiple R-squared:  0.9009, Adjusted R-squared:  0.8984 
## F-statistic: 354.7 on 12 and 468 DF,  p-value: < 2.2e-16
f_ardl92_hanover  <- forecast(mod_ardl92_hanover, 
                              x= t(full_cases_wastewater_weather_data_hanover_test[,7]),
                              h=14,
                              interval = TRUE)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_ardl92_hanover$forecasts[,2])
## [1] 0.3413698
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_ardl92_hanover$forecasts[,2])
## [1] 0.2535276
checkresiduals(mod_ardl92_hanover)
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -0.2937247715  0.0725525051  0.2357132291  0.0428823729  0.2024050180 
##            15            16            17            18            19 
##  0.1383610625 -0.2676940703 -0.0134268985 -0.0117135696  0.0764144973 
##            20            21            22            23            24 
##  0.1981303419  0.2113455282  0.2379975457  0.0051074184 -0.0491099861 
##            25            26            27            28            29 
## -0.1261627848 -0.2905892448  0.6591943825 -0.2276926085  0.3194060555 
##            30            31            32            33            34 
## -0.4551185306  0.1445597583 -0.0568366994 -0.0627825150  0.2938045965 
##            35            36            37            38            39 
##  0.3709020208  0.2960118006 -0.0942220382 -0.1734517743  0.0226241743 
##            40            41            42            43            44 
##  0.0270041027  0.1869636936 -0.2392370427 -0.3142958568  0.2411378355 
##            45            46            47            48            49 
## -0.0771738354 -0.1404487333 -0.3677277543  0.0431663383 -0.4316978812 
##            50            51            52            53            54 
##  0.2531292334 -0.1388129807 -0.4180790801  0.3174781199 -0.1916208740 
##            55            56            57            58            59 
##  0.1987007527  0.3663651337  0.2555273155  0.0239482523 -0.0530943676 
##            60            61            62            63            64 
##  0.1176959277  0.0485663629  0.1937034479  0.0685978034 -0.1774245281 
##            65            66            67            68            69 
## -0.4345144209  0.6653127842  0.1917715091  0.1800176281 -0.2190088185 
##            70            71            72            73            74 
##  0.4411357528  0.0434507919 -0.0425528122  0.0906556734 -0.1789678336 
##            75            76            77            78            79 
##  0.1817007557  0.4857756943  0.2444219798  0.1366642375  0.1156306111 
##            80            81            82            83            84 
##  0.4289515301  0.3184282752  0.1221898253 -0.0488892873 -0.1376011723 
##            85            86            87            88            89 
##  0.2397033861 -0.0037757774  0.2027015932 -0.3899420104 -0.0166538988 
##            90            91            92            93            94 
## -0.2101998536  0.0999245294 -0.2764622533  0.1196021546 -0.1969040255 
##            95            96            97            98            99 
##  0.4246182427  0.0866703371 -0.2692966292  0.1022241170  0.1845049143 
##           100           101           102           103           104 
##  0.2844814622  0.0567238754  0.0718908996 -0.0193711583  0.3085424980 
##           105           106           107           108           109 
## -0.2717931726 -1.1953301375  0.6730721267 -0.1633869109  0.3155728050 
##           110           111           112           113           114 
## -0.0091031336  0.3984023024  0.5548959808 -1.0566851701  0.7880259146 
##           115           116           117           118           119 
##  0.0939056670  0.1223389249  0.0491446447  0.1406920939 -0.6743811810 
##           120           121           122           123           124 
##  0.3197780047  0.2157982532 -0.5256451527  0.0489862102 -1.0499510287 
##           125           126           127           128           129 
##  0.4394807412  0.3652649330 -0.5195959156  0.3998504377 -0.1191334670 
##           130           131           132           133           134 
##  0.4433747581 -0.0947424623  0.0832120425 -0.5277903645 -0.1063871747 
##           135           136           137           138           139 
##  0.2841217598 -0.8616391162 -0.3790968311  0.7288172894  0.5071672677 
##           140           141           142           143           144 
## -0.2286002821  0.0314926421 -0.6829332184  0.9703729179 -0.7449079747 
##           145           146           147           148           149 
##  0.2679769867 -0.3756868031  0.2328898204  0.2944174911 -0.4603615280 
##           150           151           152           153           154 
## -0.2255009541 -0.0717132127 -0.1898146588  0.0234792637  0.3163682613 
##           155           156           157           158           159 
##  0.2111669824 -0.2559877586 -0.0066921052 -0.1121490904 -0.0041986795 
##           160           161           162           163           164 
##  0.0424280537  0.3866164312  0.3249911181 -0.4301015712 -0.0217016916 
##           165           166           167           168           169 
## -0.2131773712  0.0505797912  0.0663116210  0.2714930485  0.2043053345 
##           170           171           172           173           174 
## -0.3305856683  0.0199390521 -0.0987236863 -0.0879896828 -0.1770255319 
##           175           176           177           178           179 
##  0.9075754923 -0.1057388530 -0.4752423733 -0.1321610850 -0.3801484525 
##           180           181           182           183           184 
##  0.5419094734 -0.2656456412  0.1227014611  0.2842934188 -0.3350889685 
##           185           186           187           188           189 
##  0.5033548382 -0.0068548374 -0.6091154625 -0.2503721300  0.0775616818 
##           190           191           192           193           194 
##  0.8245964949 -0.3712032240 -0.6583240983  1.1415074891  0.0754030017 
##           195           196           197           198           199 
##  0.6061736284 -0.2877254417 -0.7529277730 -0.8836332442  0.7384886742 
##           200           201           202           203           204 
##  0.0897824100  0.7422410308  0.3995787787  0.7494566948  0.8276606603 
##           205           206           207           208           209 
##  0.1200761219 -0.0654303538 -0.2230863545  0.0274122666 -0.0400149964 
##           210           211           212           213           214 
##  0.0022512118  0.2936783139  0.0990719213  0.0543947860 -0.0493149906 
##           215           216           217           218           219 
## -0.0373004111  0.3833824395  0.0185383025  0.0589516846  0.0781896085 
##           220           221           222           223           224 
##  0.1379129373 -0.0118587807 -0.1957035215  0.1029201034 -0.1096721278 
##           225           226           227           228           229 
##  0.2728494624  0.2057127592 -0.1336138055  0.0636875549  0.2787792813 
##           230           231           232           233           234 
## -0.0819055576 -0.2028231242  0.1279166587  0.6104019865  0.5059773785 
##           235           236           237           238           239 
##  0.2927907643  0.3309247297 -0.2103299644 -0.6020374951 -0.2870039810 
##           240           241           242           243           244 
## -0.1242200552  0.1387655073  0.1208341292  0.1154879578 -0.1883789764 
##           245           246           247           248           249 
## -0.1700820686 -0.4658964526 -0.8623299961  0.6775253499  0.0105449122 
##           250           251           252           253           254 
## -0.0087829510  0.1708893966 -0.2762265336 -0.0240055186  0.3459538609 
##           255           256           257           258           259 
## -0.1418843551 -0.1564006393 -0.1053706580  0.1066407105 -0.1853499335 
##           260           261           262           263           264 
## -0.2487433641 -0.2333642650 -0.2727303989 -0.1687733447 -0.0466078025 
##           265           266           267           268           269 
## -0.3995324342 -0.0677488924 -0.0471187115 -0.0601087398 -0.1041610652 
##           270           271           272           273           274 
## -0.2367197351 -0.3989378404  0.2484457684 -0.1568777389 -0.7453024532 
##           275           276           277           278           279 
##  0.3555214858  0.1002671666 -0.3868654278  0.3040895309 -0.2028461658 
##           280           281           282           283           284 
##  0.1722189512 -0.2332142707 -0.0440658053 -0.3086458206 -0.0504900649 
##           285           286           287           288           289 
##  0.1012008512  0.1700433893 -0.3481999153 -0.3257685419  0.0074100728 
##           290           291           292           293           294 
##  0.0341751172 -1.4888343080 -0.7335710249 -0.5480315814  0.7716219916 
##           295           296           297           298           299 
## -0.3682446885 -0.0486605721 -0.1022491050  0.4153375971 -0.6283894824 
##           300           301           302           303           304 
##  0.2012134868 -0.5205160486 -0.1192180476 -0.0939335491  0.4307161292 
##           305           306           307           308           309 
## -0.1346345308 -0.6418514036 -0.1222579844  0.1329938855  0.1042333559 
##           310           311           312           313           314 
##  0.5206664141 -0.6846115233 -0.3201812611  0.0842644910  0.6126850050 
##           315           316           317           318           319 
##  0.5875763431 -0.4078496593  0.2353657459  0.3752687472  0.3007912347 
##           320           321           322           323           324 
##  0.0219233387 -0.5585097529  0.0920338782  0.4985910974 -0.2534512110 
##           325           326           327           328           329 
## -0.1352546575 -0.0338617618 -1.2401356127 -0.0603990799  0.6927204590 
##           330           331           332           333           334 
##  0.8101645216  0.4554625066  0.2758977507 -0.1558902425  0.2324918101 
##           335           336           337           338           339 
## -0.1571132546 -0.3554635846  0.1720243647 -0.0368614666 -0.2860883750 
##           340           341           342           343           344 
##  0.3664146703  0.3245726439 -0.8113220120  0.3094211668  0.1261280314 
##           345           346           347           348           349 
##  0.1013780624  0.5355813115 -0.2189059879  0.4203520863  0.4150589674 
##           350           351           352           353           354 
##  0.3031770151  0.1136186294  0.3525790847  0.0603945804  0.6853335007 
##           355           356           357           358           359 
##  0.3605603211 -0.5404365110 -0.6752436627  0.5970516429  0.8297212770 
##           360           361           362           363           364 
##  0.7278450588  0.5652834562  0.5630755331 -0.1175717912 -0.1903461945 
##           365           366           367           368           369 
##  0.0972404074  0.3422963065  0.3265283580  0.0821512567  0.4221394520 
##           370           371           372           373           374 
##  0.3144995798  0.0242426290 -0.0869549654  0.1292901847  0.0371106374 
##           375           376           377           378           379 
##  0.1510399797  0.1887376894  0.2498141128 -0.1476159290 -0.2584843004 
##           380           381           382           383           384 
##  0.2742931587  0.3827214873  0.2861100347  0.0320705553 -0.8687612922 
##           385           386           387           388           389 
## -0.6717371112  0.0901594979  0.3829023442 -0.0934348061 -0.0938720504 
##           390           391           392           393           394 
##  0.1243421596  0.0823670432 -0.4012690476 -0.3010281636 -0.0453070090 
##           395           396           397           398           399 
## -0.1682664417 -0.0227266962 -0.1872004916 -0.1452224339 -0.5144250145 
##           400           401           402           403           404 
## -0.1922387100 -0.0878008475 -0.0624489138 -0.2762842108  0.0024621932 
##           405           406           407           408           409 
## -0.0521510737 -0.7394339480  0.0295910486 -0.2974149703 -0.4524329524 
##           410           411           412           413           414 
## -0.0942575808 -0.2679268279 -0.1104944518 -0.5897524445  0.3438203257 
##           415           416           417           418           419 
## -0.1288860696 -0.2150139160 -0.0303903430  0.0037787687 -0.4397751792 
##           420           421           422           423           424 
## -0.6832706400 -0.4453951363 -0.1344499452 -0.8179229615 -0.4643839583 
##           425           426           427           428           429 
##  0.5094620775 -0.3647641012  0.3028159342  0.2062176388  0.1865326352 
##           430           431           432           433           434 
## -0.5719646627  0.1892242928  0.0735899417 -0.4111611070  0.2540204491 
##           435           436           437           438           439 
##  0.0494242221 -0.5904800236 -0.1388350201 -0.1684062474 -0.1357092483 
##           440           441           442           443           444 
##  0.1638572878  0.4641277285  0.9046284552 -0.5630378051 -0.2394969869 
##           445           446           447           448           449 
## -0.1976548938 -0.2692456496  0.0486816515  0.2693518653  0.1750217782 
##           450           451           452           453           454 
## -0.3212230719 -0.1564946816 -0.2735179742 -0.1829370659 -0.0562354176 
##           455           456           457           458           459 
##  0.2751055137  0.1558797044  0.1341932450 -0.3953978283  0.2066401363 
##           460           461           462           463           464 
##  0.6027321415 -0.6998549402  0.0009059039 -0.0013204327  0.0471160495 
##           465           466           467           468           469 
##  0.3756997387  0.7391826271 -0.3535968665 -0.1872812377 -0.4631693409 
##           470           471           472           473           474 
## -0.4225914438  0.2121285303  0.0406720251 -0.3903567773 -0.2813712050 
##           475           476           477           478           479 
## -0.5980437445  1.1517746897 -0.7341497203  0.2773720907  0.3208178831 
##           480           481           482           483           484 
##  0.3306062548  0.3199183622  0.5542635356 -0.8634054777 -0.1105232295 
##           485           486           487           488           489 
##  0.2386241573 -0.0339323640  0.1725353984 -0.1361991729  0.3281013347 
##           490 
## -0.0849704518

exp(f_ardl92_hanover $forecasts[1,2])
## [1] 1.518537
exp(f_ardl92_hanover $forecasts[1,1])
## [1] 0.7559629
exp(f_ardl92_hanover $forecasts[1,3])
## [1] 3.25878
exp(f_ardl92_hanover $forecasts[1,2]) - exp(full_cases_wastewater_weather_data_hanover_test[1,6])
## [1] 0.4103295
exp(f_ardl92_hanover$forecasts[7,2])
## [1] 1.893468
exp(f_ardl92_hanover$forecasts[7,1])
## [1] 0.6866424
exp(f_ardl92_hanover$forecasts[7,3])
## [1] 4.932396
exp(f_ardl92_hanover$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_hanover_test[7,6])
## [1] -0.3649397
exp(f_ardl92_hanover$forecasts[14,2])
## [1] 2.281199
exp(f_ardl92_hanover$forecasts[14,1])
## [1] 0.7101495
exp(f_ardl92_hanover$forecasts[14,3])
## [1] 7.184008
exp(f_ardl92_hanover$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_hanover_test[14,6])
## [1] 1.441972
mod_ardl144_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                              data = full_cases_wastewater_weather_data_hanover_train, 
                              p=4,q=14)
summary(mod_ardl144_hanover) #wastewater is significant at current time t
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.46038 -0.20792  0.00467  0.22007  1.13810 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -0.70777    0.23897  -2.962 0.003218 ** 
## log_viral_gene.t       0.08027    0.02945   2.725 0.006674 ** 
## log_viral_gene.1      -0.02027    0.04045  -0.501 0.616493    
## log_viral_gene.2      -0.03592    0.03996  -0.899 0.369136    
## log_viral_gene.3       0.03672    0.04051   0.906 0.365207    
## log_viral_gene.4      -0.01029    0.02996  -0.344 0.731279    
## log_mean_new_cases.1   0.42417    0.04675   9.074  < 2e-16 ***
## log_mean_new_cases.2   0.12347    0.05092   2.425 0.015700 *  
## log_mean_new_cases.3   0.03161    0.05067   0.624 0.533023    
## log_mean_new_cases.4   0.16856    0.05057   3.333 0.000928 ***
## log_mean_new_cases.5   0.11553    0.05096   2.267 0.023852 *  
## log_mean_new_cases.6   0.10487    0.05117   2.050 0.040979 *  
## log_mean_new_cases.7   0.19118    0.05140   3.719 0.000225 ***
## log_mean_new_cases.8  -0.02135    0.05163  -0.414 0.679433    
## log_mean_new_cases.9  -0.06158    0.05162  -1.193 0.233514    
## log_mean_new_cases.10  0.04080    0.05123   0.796 0.426224    
## log_mean_new_cases.11 -0.07317    0.05054  -1.448 0.148400    
## log_mean_new_cases.12 -0.17253    0.05060  -3.409 0.000709 ***
## log_mean_new_cases.13 -0.01017    0.05077  -0.200 0.841293    
## log_mean_new_cases.14  0.04810    0.04652   1.034 0.301651    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3733 on 456 degrees of freedom
## Multiple R-squared:  0.9058, Adjusted R-squared:  0.9018 
## F-statistic: 230.7 on 19 and 456 DF,  p-value: < 2.2e-16
f_ardl144_hanover  <- forecast(mod_ardl144_hanover, 
                               x= t(full_cases_wastewater_weather_data_hanover_test[,7]),
                               h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_ardl144_hanover$forecasts)
## [1] 0.3511095
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_ardl144_hanover$forecasts)
## [1] 0.2439619
checkresiduals(mod_ardl144_hanover)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
##  0.2320250133 -0.2300846268  0.0000143706  0.0349184796  0.1300112992 
##            20            21            22            23            24 
##  0.2269348984  0.1717666001  0.1929651664  0.0276352616  0.0080272764 
##            25            26            27            28            29 
## -0.0713886061 -0.2495602697  0.6592158224 -0.2817253131  0.2281100247 
##            30            31            32            33            34 
## -0.4411943580  0.1532365877 -0.0156566104 -0.0368831583  0.3346043139 
##            35            36            37            38            39 
##  0.4113480753  0.2915138018 -0.0997877747 -0.1377274602  0.1171967690 
##            40            41            42            43            44 
##  0.0728981131  0.1734009525 -0.2879403958 -0.3728671236  0.1890581729 
##            45            46            47            48            49 
## -0.1022702209 -0.0827927705 -0.2529601975  0.1074346433 -0.4642147723 
##            50            51            52            53            54 
##  0.1892439249 -0.1057709383 -0.3241490111  0.3878422125 -0.1495403100 
##            55            56            57            58            59 
##  0.1900141135  0.4615531246  0.3511700964  0.0720535093 -0.0091372757 
##            60            61            62            63            64 
##  0.1585805658  0.0627268763  0.2426052785 -0.0166373358 -0.2292515413 
##            65            66            67            68            69 
## -0.4744973309  0.6008862571  0.2073200738  0.2291873909 -0.1416975454 
##            70            71            72            73            74 
##  0.5162789631  0.0382342363 -0.0139391476  0.1244000009 -0.1324811504 
##            75            76            77            78            79 
##  0.1806257338  0.3842008718  0.1482590155  0.1888318873  0.2084414841 
##            80            81            82            83            84 
##  0.4442406183  0.2583154205  0.1566341352 -0.0632220448 -0.1780292972 
##            85            86            87            88            89 
##  0.1621321007 -0.0922429867  0.1587955484 -0.3455528342 -0.0183598240 
##            90            91            92            93            94 
## -0.2600593310  0.0838163904 -0.2549937596  0.1889810200 -0.1407017110 
##            95            96            97            98            99 
##  0.3930241521  0.1118718051 -0.2078136487  0.1732990744  0.2469364644 
##           100           101           102           103           104 
##  0.2710655412  0.0661076090  0.1018750162  0.0054691777  0.3025625885 
##           105           106           107           108           109 
## -0.2917380929 -1.2157419496  0.6787783795 -0.1665914037  0.1909557743 
##           110           111           112           113           114 
## -0.0118667727  0.4722382175  0.6537192716 -0.9720931974  0.8171383491 
##           115           116           117           118           119 
##  0.2040302630  0.1837301425 -0.0368683171 -0.0199302637 -0.6525793571 
##           120           121           122           123           124 
##  0.3254338513  0.1973811120 -0.5357641822  0.1292569313 -1.0387411914 
##           125           126           127           128           129 
##  0.2443203713  0.3406920104 -0.3873295459  0.4437870875 -0.0110202004 
##           130           131           132           133           134 
##  0.5014494882 -0.1170924538  0.1757481459 -0.3335584158 -0.0533921273 
##           135           136           137           138           139 
##  0.2212596638 -1.0139588858 -0.3790866848  0.7811903992  0.4771799052 
##           140           141           142           143           144 
## -0.2222172829  0.1247146408 -0.5937242123  0.9887303643 -0.6838390958 
##           145           146           147           148           149 
##  0.2061571359 -0.3248286803  0.2071951389  0.1451888846 -0.5530813689 
##           150           151           152           153           154 
## -0.0966200035  0.0725994322 -0.2369505679 -0.0935440009  0.3477337301 
##           155           156           157           158           159 
##  0.3082097594 -0.2227478650  0.0252603792 -0.1180233746  0.0527919414 
##           160           161           162           163           164 
##  0.0812329174  0.3045683433  0.2702736226 -0.4723402109 -0.0742506942 
##           165           166           167           168           169 
## -0.1516700072  0.1084299846  0.1099269486  0.1543529346  0.1500562454 
##           170           171           172           173           174 
## -0.3690503180 -0.0205702751 -0.0804764764 -0.0125241231 -0.1551577888 
##           175           176           177           178           179 
##  0.8063617348 -0.0780030506 -0.5291281104 -0.1372098955 -0.3568820576 
##           180           181           182           183           184 
##  0.5876690665 -0.2302114632  0.0426042038  0.1276556525 -0.3711892151 
##           185           186           187           188           189 
##  0.3821713998  0.0276806120 -0.3800235232 -0.2661621638 -0.0798252779 
##           190           191           192           193           194 
##  0.7329291365 -0.3894195176 -0.5834256981  1.1380973689  0.0705385167 
##           195           196           197           198           199 
##  0.5712203719 -0.3233007880 -0.6890275589 -0.9318087445  0.4993778499 
##           200           201           202           203           204 
## -0.1129632849  0.7596497156  0.5279075014  0.6634235694  0.7060288938 
##           205           206           207           208           209 
##  0.2728171372  0.0623981186 -0.1525779449 -0.0898095995 -0.4107703365 
##           210           211           212           213           214 
## -0.3803512438  0.1549021151  0.0686055849  0.0310103853 -0.0764072753 
##           215           216           217           218           219 
##  0.0140504236  0.4216517065 -0.0034951842 -0.0075872257  0.0454263630 
##           220           221           222           223           224 
##  0.1093738013 -0.0568475137 -0.2096177809  0.1098621148 -0.1215136008 
##           225           226           227           228           229 
##  0.2041466416  0.1491517355 -0.1573109650  0.0925667395  0.3201375162 
##           230           231           232           233           234 
## -0.0932453663 -0.1914384460  0.1150312084  0.5861666964  0.4695548625 
##           235           236           237           238           239 
##  0.2970770151  0.3010274480 -0.1672028795 -0.6178253411 -0.3075969303 
##           240           241           242           243           244 
## -0.2082358348  0.1082555938  0.0397408411 -0.0781370906 -0.1812830250 
##           245           246           247           248           249 
## -0.0715619905 -0.2950765747 -0.7991340757  0.7493444370  0.0277891587 
##           250           251           252           253           254 
## -0.1000750479  0.1961736231 -0.1532057808  0.0561568517  0.4509695586 
##           255           256           257           258           259 
## -0.0561706886 -0.1106163370 -0.1063070571 -0.0123132162 -0.2773329012 
##           260           261           262           263           264 
## -0.1739023622 -0.1372940145 -0.2603351082 -0.1856692693 -0.1100029435 
##           265           266           267           268           269 
## -0.4053952517  0.0124290030  0.0669362304 -0.0220836887 -0.0375369249 
##           270           271           272           273           274 
## -0.1419381642 -0.3607986817  0.2681801439 -0.1215579572 -0.7396969867 
##           275           276           277           278           279 
##  0.3697099511  0.1293586740 -0.4011285781  0.3328928730 -0.1165749959 
##           280           281           282           283           284 
##  0.2297345452 -0.2299800115 -0.0608823386 -0.3267949361 -0.0025010517 
##           285           286           287           288           289 
##  0.0644444196  0.0701412964 -0.2968815121 -0.2653501889 -0.0152858289 
##           290           291           292           293           294 
##  0.0558851139 -1.4603755107 -0.7648996770 -0.5913572414  0.7002986103 
##           295           296           297           298           299 
## -0.3741254843 -0.0057037782  0.0762723113  0.5929471875 -0.5500160748 
##           300           301           302           303           304 
##  0.2730759101 -0.3205410948 -0.0428166695 -0.2941196006  0.2210274224 
##           305           306           307           308           309 
## -0.1295237034 -0.5021145770 -0.0837131735  0.1156441659  0.0198313147 
##           310           311           312           313           314 
##  0.5430597700 -0.7234020154 -0.3205984478  0.0696901478  0.5436954910 
##           315           316           317           318           319 
##  0.6062876647 -0.1745014570  0.2708071071  0.2894306578  0.2210611394 
##           320           321           322           323           324 
##  0.0472606162 -0.4833797089  0.1025954041  0.3285643577 -0.5241406006 
##           325           326           327           328           329 
## -0.2634225673  0.0492345286 -1.2091492452 -0.2276069145  0.5956708340 
##           330           331           332           333           334 
##  0.8159789018  0.5102077200  0.3134650203 -0.1604034240  0.3051622425 
##           335           336           337           338           339 
## -0.0616198104 -0.3842638935  0.1145812366 -0.1926680658 -0.6270725788 
##           340           341           342           343           344 
##  0.1795970159  0.3023288005 -0.6475701299  0.3273887520  0.2354417156 
##           345           346           347           348           349 
##  0.0083160036  0.5343726003 -0.1930468830  0.4622699905  0.5427911877 
##           350           351           352           353           354 
##  0.3218424245  0.0209946864  0.3504951811  0.0281858198  0.5289258297 
##           355           356           357           358           359 
##  0.3076804536 -0.5669043519 -0.7399932806  0.4920667410  0.6732596071 
##           360           361           362           363           364 
##  0.6530432962  0.5873000138  0.6125703814 -0.1179175164 -0.2253975234 
##           365           366           367           368           369 
##  0.0649561480  0.3775535042  0.2841923641 -0.1915909893  0.1343152552 
##           370           371           372           373           374 
##  0.2501988565  0.1193298318  0.0036778451  0.2056891041  0.0434580953 
##           375           376           377           378           379 
## -0.0076332665  0.0038717271  0.2096826562 -0.0666120484 -0.2222461427 
##           380           381           382           383           384 
##  0.2688187193  0.4132895302  0.3098874051 -0.0048492087 -0.8733770573 
##           385           386           387           388           389 
## -0.6706971117  0.1095052226  0.3587255761 -0.0425208246 -0.0185968557 
##           390           391           392           393           394 
##  0.1176113655  0.0793784585 -0.2832013293 -0.1451613476  0.1382852449 
##           395           396           397           398           399 
## -0.1223831004 -0.1669238330 -0.3279819878 -0.0800620886 -0.3394125607 
##           400           401           402           403           404 
## -0.0892425084 -0.0323232762  0.0287631448 -0.2193074140 -0.0128360581 
##           405           406           407           408           409 
##  0.0108395011 -0.6106048622  0.0824733859 -0.2087651664 -0.3972848782 
##           410           411           412           413           414 
## -0.0651620460 -0.3015389795 -0.0820667767 -0.4796685020  0.4414204728 
##           415           416           417           418           419 
## -0.0343043217 -0.0857207655  0.0584775152  0.0010112751 -0.3548307299 
##           420           421           422           423           424 
## -0.6124304550 -0.4652583067 -0.1293264643 -0.8053766443 -0.5002958966 
##           425           426           427           428           429 
##  0.4688260419 -0.2401283857  0.4179923558  0.3093850326  0.3576574681 
##           430           431           432           433           434 
## -0.3958215691  0.2340110628  0.0634697767 -0.3865980684  0.2745122858 
##           435           436           437           438           439 
## -0.1218894973 -0.6588760861 -0.1129545478 -0.2027723541 -0.1701413701 
##           440           441           442           443           444 
##  0.2246033620  0.4941633337  0.8351275584 -0.4641952110 -0.1636153485 
##           445           446           447           448           449 
## -0.2019021139 -0.1736399643  0.0362359011  0.1788713562  0.0350802184 
##           450           451           452           453           454 
## -0.3625821869 -0.2257768002 -0.3004671611 -0.0152157890  0.0537056711 
##           455           456           457           458           459 
##  0.1963532056  0.0377203338  0.1115567708 -0.4189798654  0.2197457705 
##           460           461           462           463           464 
##  0.7445384771 -0.6453127672 -0.0864044489 -0.1265151657 -0.0414112259 
##           465           466           467           468           469 
##  0.2635374263  0.7002040516 -0.2519372951 -0.1744044534 -0.5277812710 
##           470           471           472           473           474 
## -0.5665540620  0.1907896541  0.0679088360 -0.5417137734 -0.4195097311 
##           475           476           477           478           479 
## -0.6749111453  1.0782110734 -0.6108349490  0.4326265163  0.3875669153 
##           480           481           482           483           484 
##  0.2650572381  0.2321079461  0.5394385073 -0.7497957365 -0.0738481523 
##           485           486           487           488           489 
##  0.1564739115 -0.2600493119  0.0135448991 -0.1512259159  0.2095058645 
##           490 
## -0.1656435051

mod_ardl113_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                              data = full_cases_wastewater_weather_data_hanover_train, 
                              p=13,q=1)
f_ardl113_hanover  <- forecast(mod_ardl113_hanover, x= t(full_cases_wastewater_weather_data_hanover_test[,7]),h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_ardl113_hanover$forecasts)
## [1] 0.4558737
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_ardl113_hanover$forecasts)
## [1] 0.3146763
checkresiduals(mod_ardl113_hanover)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
##  2.464235e-01  1.079214e-01 -3.004630e-01  1.146269e-01  1.634199e-01 
##            19            20            21            22            23 
##  2.233038e-01  2.089266e-01  1.961528e-01  2.025143e-01 -8.934657e-02 
##            24            25            26            27            28 
## -1.165070e-04  4.644103e-02 -1.229424e-01  8.704921e-01 -3.485893e-01 
##            29            30            31            32            33 
##  4.296294e-01 -5.434006e-01  4.591188e-01  2.953427e-02  7.280774e-02 
##            34            35            36            37            38 
##  4.296377e-01  4.310221e-01  1.366181e-01 -2.663207e-01  2.369030e-02 
##            39            40            41            42            43 
##  2.993593e-01  1.616120e-01  3.200558e-01 -1.097091e-01 -1.771500e-01 
##            44            45            46            47            48 
##  2.759151e-01 -5.614891e-02 -1.513982e-02 -2.425796e-01  2.745084e-01 
##            49            50            51            52            53 
## -3.950627e-01  4.078371e-01 -1.848751e-01 -3.081471e-01  4.125180e-01 
##            54            55            56            57            58 
## -2.619204e-01  3.928730e-01  2.722821e-01  1.957764e-01 -1.152038e-01 
##            59            60            61            62            63 
## -3.133690e-02  2.275582e-01  1.220873e-01  4.100802e-01  1.097825e-01 
##            64            65            66            67            68 
## -3.351586e-02 -1.332685e-01  9.242413e-01  1.932721e-01  1.574233e-01 
##            69            70            71            72            73 
## -1.507731e-01  5.909563e-01 -6.515428e-02  5.172955e-02  3.029702e-01 
##            74            75            76            77            78 
## -2.456603e-04  3.266876e-01  3.205375e-01  3.190554e-01  8.812246e-02 
##            79            80            81            82            83 
## -4.173743e-02  5.607035e-01  2.586454e-01  1.669316e-01  1.679341e-01 
##            84            85            86            87            88 
##  9.005550e-02  4.301698e-01 -2.036480e-02  4.957941e-01 -3.665758e-01 
##            89            90            91            92            93 
##  2.342454e-01 -9.673563e-02  4.142025e-01 -2.138409e-01  1.517826e-01 
##            94            95            96            97            98 
## -1.757819e-01  4.942384e-01 -2.824774e-05 -2.657460e-01  3.263593e-01 
##            99           100           101           102           103 
##  2.360464e-01  1.843038e-01  2.402811e-02  9.507516e-02  7.453924e-03 
##           104           105           106           107           108 
##  4.273160e-01 -2.532310e-01 -9.490978e-01  1.245190e+00 -2.651686e-01 
##           109           110           111           112           113 
##  4.272234e-01 -1.886624e-01  5.328153e-01  3.786976e-01 -1.321191e+00 
##           114           115           116           117           118 
##  1.371759e+00  3.410164e-02  1.787187e-01 -6.101586e-02  3.204270e-01 
##           119           120           121           122           123 
## -5.790502e-01  4.839995e-01  3.184289e-01 -4.041732e-01  1.807141e-01 
##           124           125           126           127           128 
## -9.884007e-01  9.774889e-01  2.090924e-01 -4.622352e-01  7.442776e-01 
##           129           130           131           132           133 
## -3.064776e-01  5.737385e-01 -4.162272e-01  3.752668e-01 -5.122829e-01 
##           134           135           136           137           138 
##  5.545539e-02  5.186287e-01 -9.418320e-01 -2.106306e-03  8.818772e-01 
##           139           140           141           142           143 
##  1.984605e-01 -5.242476e-01  3.891965e-02 -5.926545e-01  1.123021e+00 
##           144           145           146           147           148 
## -1.052675e+00  6.892360e-01 -4.172753e-01  2.450462e-01  1.571643e-01 
##           149           150           151           152           153 
## -3.432682e-01  1.542178e-02  4.661517e-02 -1.400209e-01  8.230218e-02 
##           154           155           156           157           158 
##  2.741594e-01  5.706222e-02 -5.145570e-01  6.245565e-02 -7.761043e-02 
##           159           160           161           162           163 
##  1.199057e-01  3.838372e-02  5.091085e-01  1.678180e-01 -8.170937e-01 
##           164           165           166           167           168 
##  1.118598e-01 -1.518930e-01  1.779649e-01  9.953022e-02  2.616328e-01 
##           169           170           171           172           173 
##  1.053195e-01 -3.941159e-01  9.659943e-03  8.265184e-02  9.181983e-03 
##           174           175           176           177           178 
## -1.691638e-01  8.420499e-01 -4.279412e-01 -5.868518e-01 -2.490959e-01 
##           179           180           181           182           183 
## -2.433995e-01  6.580741e-01 -4.743333e-01  1.144835e-01  1.151714e-01 
##           184           185           186           187           188 
## -5.545354e-01  5.834953e-01 -2.632818e-01 -4.661061e-01 -2.926448e-01 
##           189           190           191           192           193 
##  1.405434e-01  6.479839e-01 -9.255004e-01 -5.008435e-01  1.184592e+00 
##           194           195           196           197           198 
## -3.517067e-01  4.503288e-01 -8.232858e-01 -4.901369e-01 -8.238246e-01 
##           199           200           201           202           203 
##  8.034251e-01  3.499138e-02  5.298002e-01 -1.900180e-01  3.388748e-01 
##           204           205           206           207           208 
##  3.008035e-01 -2.647209e-01  5.605422e-02 -1.561656e-01  2.706322e-01 
##           209           210           211           212           213 
## -6.116054e-02 -1.520668e-02  2.902518e-01 -7.375718e-02 -1.946204e-02 
##           214           215           216           217           218 
## -1.101271e-01  5.860545e-02  3.933577e-01 -2.067138e-01  6.567620e-02 
##           219           220           221           222           223 
##  2.240577e-02  1.117048e-01 -1.444287e-01 -1.224293e-01  2.235235e-01 
##           224           225           226           227           228 
## -1.992278e-01  2.591305e-01  9.701836e-02 -1.741103e-01  7.201539e-03 
##           229           230           231           232           233 
##  2.818788e-01 -1.599871e-01 -2.520877e-01  1.892706e-01  6.324726e-01 
##           234           235           236           237           238 
##  3.566457e-01  5.142079e-02  2.829246e-01 -2.928293e-01 -4.235424e-01 
##           239           240           241           242           243 
##  1.088054e-01  2.086854e-01  2.177394e-01  2.340707e-01 -4.925637e-02 
##           244           245           246           247           248 
## -1.978960e-01 -3.000210e-02 -1.177888e-01 -5.652781e-01  7.513701e-01 
##           249           250           251           252           253 
## -3.479821e-02 -2.405153e-01  4.745448e-02 -2.752346e-01  1.505968e-01 
##           254           255           256           257           258 
##  3.619144e-01 -1.689926e-01 -1.230290e-01 -1.604969e-01  1.582474e-01 
##           259           260           261           262           263 
## -1.624001e-01 -2.088017e-01 -3.651384e-02 -3.797311e-02 -9.440940e-02 
##           264           265           266           267           268 
##  9.062343e-03 -3.950558e-01  4.795182e-02 -1.088870e-01 -9.442184e-02 
##           269           270           271           272           273 
## -1.839960e-01 -2.549597e-01 -3.852999e-01  3.209049e-01 -3.319555e-01 
##           274           275           276           277           278 
## -7.272679e-01  5.069109e-01 -1.457574e-01 -5.504818e-01  2.266118e-01 
##           279           280           281           282           283 
## -3.179506e-01  2.094433e-01 -5.519211e-01  1.078107e-01 -1.970213e-01 
##           284           285           286           287           288 
## -5.879788e-02  9.916075e-02  6.117743e-02 -4.252216e-01 -3.849268e-01 
##           289           290           291           292           293 
##  8.777387e-02  1.012035e-01 -1.544862e+00 -2.378311e-01 -2.628968e-01 
##           294           295           296           297           298 
##  9.015871e-01 -8.794420e-01 -1.230085e-01 -3.705736e-01  1.706511e-01 
##           299           300           301           302           303 
## -1.020351e+00  3.866089e-01 -5.547021e-01 -1.056843e-01 -3.334892e-01 
##           304           305           306           307           308 
##  3.164453e-01 -4.360064e-01 -8.882716e-01  2.548828e-02  2.294604e-01 
##           309           310           311           312           313 
## -1.025422e-01  4.121399e-01 -7.206442e-01 -1.777182e-01  4.246002e-02 
##           314           315           316           317           318 
##  5.876482e-01  5.512028e-01 -8.205486e-01  2.313312e-01  1.723283e-01 
##           319           320           321           322           323 
##  1.575748e-01 -7.467539e-02 -4.518359e-01  1.045983e-01  4.637128e-01 
##           324           325           326           327           328 
## -4.974163e-01 -3.237650e-01 -8.134881e-02 -1.279873e+00  2.623930e-01 
##           329           330           331           332           333 
##  6.861806e-01  6.039959e-01 -1.893753e-01 -2.304746e-01 -3.635442e-01 
##           334           335           336           337           338 
##  1.131718e-01 -1.675207e-01 -6.220534e-02  3.617296e-01 -1.905118e-01 
##           339           340           341           342           343 
## -3.644422e-01  5.003083e-01  1.981548e-01 -8.621920e-01  7.472343e-01 
##           344           345           346           347           348 
##  2.326173e-01 -9.845740e-03 -3.949285e-02 -1.092496e-01  3.196866e-01 
##           349           350           351           352           353 
##  1.918624e-01  1.899830e-01  2.726169e-01  3.127908e-01 -3.066221e-01 
##           354           355           356           357           358 
##  5.910136e-01  1.965529e-01 -6.071190e-01 -4.901459e-01  9.571559e-01 
##           359           360           361           362           363 
##  7.785486e-01  4.297194e-01  2.421168e-01  3.839082e-01 -4.124742e-01 
##           364           365           366           367           368 
## -1.594592e-01  5.397143e-01  6.824647e-01  3.062328e-01 -9.025481e-02 
##           369           370           371           372           373 
##  4.896963e-01  1.661832e-01 -1.106755e-01  1.151560e-01  4.256767e-01 
##           374           375           376           377           378 
##  1.514870e-01  1.344179e-01  2.897561e-01  3.059985e-01 -2.154945e-01 
##           379           380           381           382           383 
## -1.121419e-01  5.715364e-01  5.037337e-01  2.597667e-01  2.678688e-03 
##           384           385           386           387           388 
## -7.643494e-01 -2.123807e-01  5.206294e-01  6.137937e-01 -1.848080e-01 
##           389           390           391           392           393 
## -5.785141e-02  7.055641e-02  1.484102e-02 -3.839321e-01  9.331012e-02 
##           394           395           396           397           398 
##  3.080681e-01 -1.100519e-01  1.106134e-02 -1.627662e-01 -5.214472e-02 
##           399           400           401           402           403 
## -3.746169e-01  1.083442e-02  8.842576e-02  3.438985e-02 -2.792617e-01 
##           404           405           406           407           408 
##  3.434635e-02 -5.540646e-02 -7.308652e-01  2.899233e-01 -2.045589e-01 
##           409           410           411           412           413 
## -2.821170e-01 -6.731756e-03 -2.594225e-01 -6.360104e-02 -5.690875e-01 
##           414           415           416           417           418 
##  5.139666e-01 -2.785183e-01 -2.416842e-01  2.851707e-03 -5.649891e-02 
##           419           420           421           422           423 
## -4.842228e-01 -5.641338e-01 -1.314034e-01  5.936359e-02 -8.771841e-01 
##           424           425           426           427           428 
## -2.356548e-01  5.635040e-01 -7.011404e-01  2.578767e-01 -9.083970e-02 
##           429           430           431           432           433 
##  8.442828e-02 -8.394920e-01  2.948416e-01  9.040904e-02 -5.392915e-01 
##           434           435           436           437           438 
##  3.089598e-01 -1.191922e-01 -6.222080e-01  6.719370e-02 -1.150110e-01 
##           439           440           441           442           443 
##  3.532734e-02  4.811247e-02  4.837361e-01  5.586254e-01 -1.049955e+00 
##           444           445           446           447           448 
##  4.864179e-02 -6.837657e-02  1.778991e-03  1.089735e-01  2.395360e-01 
##           449           450           451           452           453 
##  3.266427e-03 -6.188671e-01  1.802997e-03 -3.534254e-01  1.084297e-01 
##           454           455           456           457           458 
## -8.825934e-02  6.616639e-02 -1.929277e-01 -1.725775e-01 -4.585026e-01 
##           459           460           461           462           463 
##  1.243172e-01  6.099323e-01 -1.027856e+00 -3.668340e-02 -1.753207e-01 
##           464           465           466           467           468 
## -1.369623e-03  2.937642e-01  4.236799e-01 -5.605061e-01 -4.239888e-01 
##           469           470           471           472           473 
## -6.300182e-01 -2.718373e-01  3.205351e-01  4.337715e-02 -6.084656e-01 
##           474           475           476           477           478 
## -3.567059e-01 -7.152771e-01  1.130024e+00 -1.263828e+00  4.330856e-01 
##           479           480           481           482           483 
## -2.140565e-02  4.015963e-02 -1.016249e-01  2.597769e-01 -1.023427e+00 
##           484           485           486           487           488 
##  1.045923e-02  2.361395e-01  1.646838e-02  4.290089e-02 -3.014100e-01 
##           489           490 
##  3.222930e-01 -3.431487e-01

mod_ardl51_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene,
                               data = full_cases_wastewater_weather_data_hanover_train, 
                               p=1,q=5)
f_ardl51_hanover  <- forecast(mod_ardl51_hanover, x= t(full_cases_wastewater_weather_data_hanover_test[,7]),h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_ardl51_hanover$forecasts)
## [1] 0.3492377
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_ardl51_hanover$forecasts)
## [1] 0.2607468
checkresiduals(mod_ardl51_hanover)
## Time Series:
## Start = 6 
## End = 490 
## Frequency = 1 
##             6             7             8             9            10 
##  0.0328220635  0.0763639764  0.0046540748 -0.2435933565 -0.2577974163 
##            11            12            13            14            15 
##  0.0516759042  0.2088724684  0.0514321076  0.2379440878  0.1355477060 
##            16            17            18            19            20 
## -0.3404018235 -0.0762795959  0.0181905028  0.1338161858  0.2168810809 
##            21            22            23            24            25 
##  0.2438434500  0.2303942844 -0.0762856882 -0.0914221855 -0.1075191351 
##            26            27            28            29            30 
## -0.2528528180  0.6965248952 -0.1966205776  0.3446781039 -0.4866767653 
##            31            32            33            34            35 
##  0.1234975029 -0.1079042044 -0.0359000648  0.3911615870  0.3984112673 
##            36            37            38            39            40 
##  0.2589576236 -0.1825436274 -0.2006553270  0.0230820382  0.0343408531 
##            41            42            43            44            45 
##  0.2670376906 -0.1516682173 -0.2924471169  0.1663005442 -0.1483611933 
##            46            47            48            49            50 
## -0.1304079347 -0.3038889904  0.1123624375 -0.4812507250  0.2015047254 
##            51            52            53            54            55 
## -0.1167572451 -0.3905704203  0.2873605917 -0.2100333373  0.1991205844 
##            56            57            58            59            60 
##  0.3377884608  0.2878576352 -0.0112247739 -0.1244478406  0.1350154484 
##            61            62            63            64            65 
##  0.0195464901  0.2220958294  0.1528199852 -0.1264946905 -0.4535805420 
##            66            67            68            69            70 
##  0.6484584361  0.2054829326  0.2050423000 -0.1984259567  0.4654973111 
##            71            72            73            74            75 
## -0.0671907248 -0.1283758975  0.1797225951 -0.0835458903  0.1659951176 
##            76            77            78            79            80 
##  0.4490215584  0.2860305338  0.1319700286  0.0588109250  0.4287021695 
##            81            82            83            84            85 
##  0.2943369054  0.1532665521  0.0345187105 -0.0896816612  0.2144690016 
##            86            87            88            89            90 
## -0.0330599322  0.2560337493 -0.3321519426  0.0162492878 -0.2455974734 
##            91            92            93            94            95 
##  0.0801445848 -0.2584900654  0.1513020956 -0.1939918082  0.3837230463 
##            96            97            98            99           100 
##  0.0547884287 -0.2599072375  0.1123651047  0.1804603978  0.2614960953 
##           101           102           103           104           105 
##  0.0424809710  0.1340920949 -0.0143607607  0.2352550159 -0.2827483060 
##           106           107           108           109           110 
## -1.1302809975  0.7432318166 -0.1589489130  0.3097466678  0.0061081782 
##           111           112           113           114           115 
##  0.4755997916  0.4218489023 -1.2412130845  0.8894642542  0.1736875469 
##           116           117           118           119           120 
##  0.1196716528  0.0347390041  0.2573019650 -0.6944435121  0.1391592637 
##           121           122           123           124           125 
##  0.3016683552 -0.3959176803  0.0386666617 -1.0500290453  0.4497152879 
##           126           127           128           129           130 
##  0.2709627709 -0.4689198507  0.5111916380 -0.1142953724  0.3496322443 
##           131           132           133           134           135 
## -0.2676530964  0.1659780716 -0.4463626784 -0.1615703017  0.3017713966 
##           136           137           138           139           140 
## -0.8354819509 -0.3468889952  0.7393470192  0.4974614391 -0.3100390034 
##           141           142           143           144           145 
##  0.0453809400 -0.6179847621  0.8259374456 -0.8408228200  0.4631916712 
##           146           147           148           149           150 
## -0.2557795324  0.1866542614  0.1614221641 -0.4652713135 -0.1117735812 
##           151           152           153           154           155 
## -0.0897062933 -0.1924181773  0.0226829068  0.3536784389  0.2363930413 
##           156           157           158           159           160 
## -0.3703972094 -0.0597533758 -0.1111394618 -0.0085813883  0.0810836576 
##           161           162           163           164           165 
##  0.4954513349  0.3539532550 -0.5172433285 -0.0772402324 -0.2123895983 
##           166           167           168           169           170 
##  0.0412792389  0.0998877245  0.3862955862  0.2373448586 -0.4322589371 
##           171           172           173           174           175 
## -0.0303939708 -0.0931697330 -0.0706835758 -0.1114414152  0.9950390355 
##           176           177           178           179           180 
## -0.1180735463 -0.5891843459 -0.1791672014 -0.3778223779  0.5000949208 
##           181           182           183           184           185 
## -0.2403171841  0.3304621890  0.2767205452 -0.4924510443  0.4477556802 
##           186           187           188           189           190 
##  0.0338469806 -0.5087339694 -0.2582943953  0.0924207559  0.8144005047 
##           191           192           193           194           195 
## -0.4694046635 -0.5870480959  1.1817338961 -0.0663642675  0.5162535719 
##           196           197           198           199           200 
## -0.1963151678 -0.5703242741 -1.0192284376  0.6393847394  0.2537494702 
##           201           202           203           204           205 
##  0.8574209983  0.4158236599  0.6861549014  0.5641520776 -0.0556441121 
##           206           207           208           209           210 
##  0.0430313653 -0.0584692632  0.1210961054 -0.0146338842  0.0663894756 
##           211           212           213           214           215 
##  0.3647957113  0.0620367654  0.0126981338 -0.0629523636 -0.0214430243 
##           216           217           218           219           220 
##  0.3739499841  0.0210694328  0.1062839461  0.0850358761  0.1175722167 
##           221           222           223           224           225 
## -0.0533164168 -0.1874940075  0.1660401040 -0.0985299622  0.2591326220 
##           226           227           228           229           230 
##  0.2117532430 -0.1059687208  0.0377909043  0.2411376445 -0.0961667197 
##           231           232           233           234           235 
## -0.2048006383  0.1738253734  0.6435909589  0.4480211087  0.2890974155 
##           236           237           238           239           240 
##  0.3713733345 -0.2365013068 -0.6876393609 -0.2439062348  0.0227495601 
##           241           242           243           244           245 
##  0.1902827515  0.1370390454  0.1523487339 -0.2253344260 -0.2996517418 
##           246           247           248           249           250 
## -0.4934033040 -0.7875744072  0.7399061986  0.0190299083 -0.0122580201 
##           251           252           253           254           255 
##  0.1331211585 -0.3163738216 -0.1568586438  0.2728530752  0.0084326031 
##           256           257           258           259           260 
## -0.0769932479 -0.1323585306  0.0766517618 -0.2622427898 -0.2515168055 
##           261           262           263           264           265 
## -0.1449868382 -0.2321509216 -0.1946487051 -0.0551976273 -0.3729364222 
##           266           267           268           269           270 
## -0.0884180350 -0.0865935053 -0.0740020480 -0.1334970730 -0.2459930128 
##           271           272           273           274           275 
## -0.4058163914  0.2160045901 -0.1650742124 -0.7230092920  0.3788789122 
##           276           277           278           279           280 
##  0.0929187302 -0.4541820485  0.2670271838 -0.1430614311  0.1394456484 
##           281           282           283           284           285 
## -0.3536265863  0.0412371834 -0.2532950550 -0.0909616854  0.1136030094 
##           286           287           288           289           290 
##  0.1921190592 -0.3576088467 -0.3550758314 -0.0036586868  0.0114470325 
##           291           292           293           294           295 
## -1.5021921163 -0.6387346274 -0.4675440103  0.7314566640 -0.4452319213 
##           296           297           298           299           300 
##  0.0519783156 -0.1029909175  0.1687274339 -0.8368689202  0.2881969954 
##           301           302           303           304           305 
## -0.3417564521 -0.1160738439 -0.1781694699  0.4635169687 -0.1511082997 
##           306           307           308           309           310 
## -0.7198262187 -0.1252055853  0.1585814134  0.0692909369  0.5557826817 
##           311           312           313           314           315 
## -0.5724581013 -0.3128230661 -0.0608994764  0.5918868466  0.6246861981 
##           316           317           318           319           320 
## -0.3289664769  0.2875034611  0.2292978657  0.1256699915  0.0191685548 
##           321           322           323           324           325 
## -0.3565629427  0.2074376511  0.3895778871 -0.2960922985 -0.0717127981 
##           326           327           328           329           330 
##  0.0179695289 -1.2714008365 -0.1175043675  0.7699542761  0.9086689033 
##           331           332           333           334           335 
##  0.3913370882  0.2526266045 -0.2293985609 -0.0127086489 -0.2079899666 
##           336           337           338           339           340 
## -0.1131232513  0.3838826059  0.0001213912 -0.3870008513  0.3150498318 
##           341           342           343           344           345 
##  0.3355352127 -0.7807368899  0.3350029154  0.2243567794  0.0781485534 
##           346           347           348           349           350 
##  0.3951910501 -0.2000113618  0.4411837609  0.3018906571  0.3190622020 
##           351           352           353           354           355 
##  0.1619486336  0.3764302200  0.0607116638  0.6370763030  0.3811565810 
##           356           357           358           359           360 
## -0.4795675581 -0.6546619406  0.6235711231  0.8334980436  0.7380326679 
##           361           362           363           364           365 
##  0.6604429889  0.6014188033 -0.3349478513 -0.3809135054  0.2144427547 
##           366           367           368           369           370 
##  0.5667367625  0.4401506055  0.1364290666  0.4303700782  0.1739142587 
##           371           372           373           374           375 
## -0.1113166631 -0.0391171104  0.2789485599  0.0984224275  0.1291849204 
##           376           377           378           379           380 
##  0.2113764973  0.2605952310 -0.2045349577 -0.2756622538  0.3269035614 
##           381           382           383           384           385 
##  0.4038043635  0.2843100537  0.0634219566 -0.8326422316 -0.7345135525 
##           386           387           388           389           390 
##  0.0358368324  0.4625133743  0.0124788708 -0.0262299298  0.0809615424 
##           391           392           393           394           395 
## -0.1336994032 -0.5539679182 -0.1992215662  0.1302209005 -0.1510501884 
##           396           397           398           399           400 
## -0.0639693978 -0.1713004899 -0.1639580458 -0.6105724450 -0.1975370198 
##           401           402           403           404           405 
## -0.0214919389 -0.0493713098 -0.2997190318 -0.0023490470 -0.0974532642 
##           406           407           408           409           410 
## -0.8224190622  0.0266993747 -0.2308880999 -0.4371940364 -0.1164118077 
##           411           412           413           414           415 
## -0.2275735298 -0.1308310735 -0.6980065485  0.3727645512 -0.1157955898 
##           416           417           418           419           420 
## -0.2737999656 -0.0531834707  0.0192993999 -0.4943486455 -0.7445203183 
##           421           422           423           424           425 
## -0.3543762177 -0.0757218418 -0.8671141109 -0.4287120683  0.5694277034 
##           426           427           428           429           430 
## -0.4445397746  0.1713955501  0.2027104232  0.2147761718 -0.7227408330 
##           431           432           433           434           435 
##  0.1546070989  0.1795745860 -0.4470909945  0.2458108177  0.1368398795 
##           436           437           438           439           440 
## -0.5883129073 -0.2376174611 -0.1186835791 -0.0603621476  0.1022441687 
##           441           442           443           444           445 
##  0.5087521282  0.9328888879 -0.6850991824 -0.2844535286 -0.1935423103 
##           446           447           448           449           450 
## -0.2625154669  0.0435433877  0.4215548170  0.2955986368 -0.4687384889 
##           451           452           453           454           455 
## -0.2531259777 -0.2658642119 -0.1838665549 -0.0307415923  0.3798346395 
##           456           457           458           459           460 
##  0.1777119377  0.0195442935 -0.4597632990  0.2016805926  0.5829627307 
##           461           462           463           464           465 
## -0.7078806473  0.0767206788  0.0507332791  0.0002578649  0.2928540310 
##           466           467           468           469           470 
##  0.8311943554 -0.2669171559 -0.3177463592 -0.5377698565 -0.3955878865 
##           471           472           473           474           475 
##  0.2222802539  0.1631996470 -0.2452567405 -0.3212481852 -0.7211024806 
##           476           477           478           479           480 
##  1.0562259546 -0.7695242988  0.4030654240  0.3858950782  0.2753018234 
##           481           482           483           484           485 
##  0.1305945402  0.5064635684 -0.7354071492 -0.1479216905  0.2244361081 
##           486           487           488           489           490 
##  0.0604289165  0.2062820948 -0.0692500089  0.3670844588 -0.2341416187

lowest_rmse_weather_hanover <- Inf
best_mod_weather_hanover <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    remove <- list(p =list(mean_precipation = c(1:p),
                           mean_temp= c(1:p)))
    mod <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                     mean_temp,data = full_cases_wastewater_weather_data_hanover_train, 
                   p=p,q=q,
                   remove = remove)
    f <- forecast(mod, 
                  x= t(full_cases_wastewater_weather_data_hanover_test[,c(7,4,5)]),
                  h=14)
    forecast_acc <- rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
                         f$forecasts) #interchanged between RMSE and MAE
    if (forecast_acc<lowest_rmse_weather_hanover){
      lowest_rmse_weather_hanover <- forecast_acc
      best_mod_weather_hanover <- mod 
    }
  }
}

lowest_rmse_weather_hanover #0.35
## [1] 0.3558231
summary(best_mod_weather_hanover) #ARDL(9,14) (lowest RMSE), ARDL(14,14) (lowest MAE)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.44868 -0.20312 -0.00174  0.24075  1.06440 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.4196023  0.2826546  -1.485  0.13838    
## log_viral_gene.t      0.0899460  0.0306394   2.936  0.00350 ** 
## log_viral_gene.1     -0.0165224  0.0415725  -0.397  0.69123    
## log_viral_gene.2     -0.0498911  0.0415549  -1.201  0.23054    
## log_viral_gene.3      0.0439677  0.0423291   1.039  0.29950    
## log_viral_gene.4      0.0578390  0.0424261   1.363  0.17347    
## log_viral_gene.5     -0.0494551  0.0427692  -1.156  0.24816    
## log_viral_gene.6      0.0005646  0.0428454   0.013  0.98949    
## log_viral_gene.7     -0.0275667  0.0428398  -0.643  0.52024    
## log_viral_gene.8      0.0607833  0.0428001   1.420  0.15625    
## log_viral_gene.9     -0.0885477  0.0427327  -2.072  0.03882 *  
## log_viral_gene.10     0.0100525  0.0425001   0.237  0.81313    
## log_viral_gene.11     0.0635569  0.0425085   1.495  0.13558    
## log_viral_gene.12    -0.0298949  0.0419159  -0.713  0.47609    
## log_viral_gene.13    -0.0213158  0.0419694  -0.508  0.61178    
## log_viral_gene.14    -0.0186732  0.0308532  -0.605  0.54533    
## mean_precipation.t   -0.0681444  0.0428634  -1.590  0.11258    
## mean_temp.t           0.0012639  0.0014208   0.890  0.37417    
## log_mean_new_cases.1  0.4409079  0.0470515   9.371  < 2e-16 ***
## log_mean_new_cases.2  0.1235043  0.0515498   2.396  0.01699 *  
## log_mean_new_cases.3  0.0343170  0.0513172   0.669  0.50402    
## log_mean_new_cases.4  0.1509435  0.0511587   2.950  0.00334 ** 
## log_mean_new_cases.5  0.0836452  0.0513815   1.628  0.10424    
## log_mean_new_cases.6  0.0743482  0.0509181   1.460  0.14495    
## log_mean_new_cases.7  0.1655124  0.0510728   3.241  0.00128 ** 
## log_mean_new_cases.8 -0.0494097  0.0515274  -0.959  0.33812    
## log_mean_new_cases.9 -0.0722661  0.0471558  -1.532  0.12610    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3781 on 449 degrees of freedom
## Multiple R-squared:  0.9048, Adjusted R-squared:  0.8993 
## F-statistic: 164.1 on 26 and 449 DF,  p-value: < 2.2e-16
remove <- list(p =list(mean_precipation = c(1:14),
                       mean_temp= c(1:14)))
mod_ardl914_weather_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                 mean_temp,data = full_cases_wastewater_weather_data_hanover_train, 
               p=14,q=9,
               remove = remove)
summary(mod_ardl914_weather_hanover)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.44868 -0.20312 -0.00174  0.24075  1.06440 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.4196023  0.2826546  -1.485  0.13838    
## log_viral_gene.t      0.0899460  0.0306394   2.936  0.00350 ** 
## log_viral_gene.1     -0.0165224  0.0415725  -0.397  0.69123    
## log_viral_gene.2     -0.0498911  0.0415549  -1.201  0.23054    
## log_viral_gene.3      0.0439677  0.0423291   1.039  0.29950    
## log_viral_gene.4      0.0578390  0.0424261   1.363  0.17347    
## log_viral_gene.5     -0.0494551  0.0427692  -1.156  0.24816    
## log_viral_gene.6      0.0005646  0.0428454   0.013  0.98949    
## log_viral_gene.7     -0.0275667  0.0428398  -0.643  0.52024    
## log_viral_gene.8      0.0607833  0.0428001   1.420  0.15625    
## log_viral_gene.9     -0.0885477  0.0427327  -2.072  0.03882 *  
## log_viral_gene.10     0.0100525  0.0425001   0.237  0.81313    
## log_viral_gene.11     0.0635569  0.0425085   1.495  0.13558    
## log_viral_gene.12    -0.0298949  0.0419159  -0.713  0.47609    
## log_viral_gene.13    -0.0213158  0.0419694  -0.508  0.61178    
## log_viral_gene.14    -0.0186732  0.0308532  -0.605  0.54533    
## mean_precipation.t   -0.0681444  0.0428634  -1.590  0.11258    
## mean_temp.t           0.0012639  0.0014208   0.890  0.37417    
## log_mean_new_cases.1  0.4409079  0.0470515   9.371  < 2e-16 ***
## log_mean_new_cases.2  0.1235043  0.0515498   2.396  0.01699 *  
## log_mean_new_cases.3  0.0343170  0.0513172   0.669  0.50402    
## log_mean_new_cases.4  0.1509435  0.0511587   2.950  0.00334 ** 
## log_mean_new_cases.5  0.0836452  0.0513815   1.628  0.10424    
## log_mean_new_cases.6  0.0743482  0.0509181   1.460  0.14495    
## log_mean_new_cases.7  0.1655124  0.0510728   3.241  0.00128 ** 
## log_mean_new_cases.8 -0.0494097  0.0515274  -0.959  0.33812    
## log_mean_new_cases.9 -0.0722661  0.0471558  -1.532  0.12610    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3781 on 449 degrees of freedom
## Multiple R-squared:  0.9048, Adjusted R-squared:  0.8993 
## F-statistic: 164.1 on 26 and 449 DF,  p-value: < 2.2e-16
f_ardl914_weather_hanover <- forecast(mod_ardl914_weather_hanover, 
                                      x= t(full_cases_wastewater_weather_data_hanover_test[,c(7,4,5)]),
                                      h=14, interval = TRUE)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
     f_ardl914_weather_hanover$forecasts) 
## [1] NA
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
     f_ardl914_weather_hanover$forecasts) 
## [1] NA
checkresiduals(mod_ardl914_weather_hanover)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
##  0.1448596577 -0.2701201876 -0.0360931405 -0.0023485448  0.0614578988 
##            20            21            22            23            24 
##  0.1902709444  0.2167612468  0.2534180814  0.0124906080 -0.0673391373 
##            25            26            27            28            29 
## -0.0572702390 -0.2956707576  0.6586152219 -0.1967724881  0.3798786680 
##            30            31            32            33            34 
## -0.4124229228  0.1839462314 -0.0076684908 -0.0769000060  0.2646881644 
##            35            36            37            38            39 
##  0.5610365432  0.2712477509 -0.1262058115 -0.1521341269  0.0523926079 
##            40            41            42            43            44 
##  0.0146428507  0.2019169199 -0.1244961774 -0.2311368899  0.1351136637 
##            45            46            47            48            49 
## -0.1218037800 -0.1837856990 -0.3071713989  0.0614630997 -0.4039120906 
##            50            51            52            53            54 
##  0.2462180252 -0.1617782398 -0.4308241779  0.2552498170 -0.1901251331 
##            55            56            57            58            59 
##  0.2568329735  0.4013289783  0.2721695003  0.0276153111  0.0247032832 
##            60            61            62            63            64 
##  0.1178572409  0.0793948964  0.2863596282  0.0604595987 -0.1192258153 
##            65            66            67            68            69 
## -0.2925498100  0.6541275645  0.2461047892  0.1015965807 -0.1803475905 
##            70            71            72            73            74 
##  0.3982735025 -0.0515859099 -0.0712090424  0.1070034088 -0.2364975135 
##            75            76            77            78            79 
##  0.0917511945  0.3081202359  0.2494557351  0.1238129916 -0.0219880744 
##            80            81            82            83            84 
##  0.3699282197  0.2198207279  0.0853133893  0.0101973100 -0.0981652886 
##            85            86            87            88            89 
##  0.2428157130 -0.1147516223  0.2060508846 -0.4271974859 -0.1076335006 
##            90            91            92            93            94 
## -0.2607917807  0.1505724942 -0.3336551074 -0.0960079920 -0.3299922517 
##            95            96            97            98            99 
##  0.2729659022  0.0323823250 -0.3264036153  0.1467719366  0.2038252861 
##           100           101           102           103           104 
##  0.1678817368  0.0567068150  0.0143095019 -0.0332827477  0.3534076498 
##           105           106           107           108           109 
## -0.2516006641 -1.1966543723  0.6407796182 -0.2063580231  0.2571231484 
##           110           111           112           113           114 
## -0.0332380415  0.3990948958  0.5476541393 -1.0538441885  0.7476816688 
##           115           116           117           118           119 
##  0.1201529499  0.0390908815 -0.0044770235  0.1391930242 -0.6692331598 
##           120           121           122           123           124 
##  0.2748240303  0.1093519383 -0.5776323059 -0.0311510605 -1.0608884452 
##           125           126           127           128           129 
##  0.4334959694  0.2732316818 -0.5354295434  0.5015650908 -0.1275566833 
##           130           131           132           133           134 
##  0.5014545010 -0.1436873305  0.1070813564 -0.5769546261 -0.2082160272 
##           135           136           137           138           139 
##  0.2809632949 -0.9246196166 -0.4340405494  0.6755398314  0.3328607260 
##           140           141           142           143           144 
## -0.3157896644  0.0061258138 -0.6882871567  0.8801283126 -0.7417254031 
##           145           146           147           148           149 
##  0.2313981319 -0.3587580249  0.1317677939  0.2569225816 -0.2974444378 
##           150           151           152           153           154 
## -0.2526086093  0.0204295192 -0.0377821688  0.0458331350  0.2584724319 
##           155           156           157           158           159 
##  0.1425418504 -0.3248379644 -0.0432838470 -0.1958435776 -0.0355487532 
##           160           161           162           163           164 
##  0.0033981643  0.4994066310  0.3561140463 -0.5568123929  0.0186486862 
##           165           166           167           168           169 
## -0.2286895116 -0.0056299829  0.0449214538  0.2374491703  0.2088266533 
##           170           171           172           173           174 
## -0.2500962338 -0.0716556393 -0.0368066937 -0.0903040122 -0.2338913764 
##           175           176           177           178           179 
##  0.7281559925 -0.0986996855 -0.4739481563 -0.3494835798 -0.4689635479 
##           180           181           182           183           184 
##  0.4176889498 -0.2120496643 -0.0484133742  0.2232366132 -0.3127062640 
##           185           186           187           188           189 
##  0.5201793630 -0.0541719126 -0.4684923147 -0.2943154516  0.0311312426 
##           190           191           192           193           194 
##  0.7069826820 -0.5017134651 -0.6043477636  1.0644039499  0.0676783365 
##           195           196           197           198           199 
##  0.5785212979 -0.4417657714 -0.6833519080 -0.7905143085  0.5643392885 
##           200           201           202           203           204 
##  0.0608334090  0.7324729182  0.3843340185  0.7305501506  0.8376321903 
##           205           206           207           208           209 
##  0.1675239318  0.0003303264 -0.2505294583  0.0558129761 -0.0566150012 
##           210           211           212           213           214 
## -0.0788039326  0.2518297493  0.0976738023  0.2603057210 -0.0747493020 
##           215           216           217           218           219 
## -0.0105319785  0.4133681646  0.0315770921  0.0036905159  0.0403785431 
##           220           221           222           223           224 
##  0.1059570929 -0.1024301168 -0.1827900433  0.0808330295 -0.1861427869 
##           225           226           227           228           229 
##  0.2023718001  0.1818968531 -0.0489599644  0.0123491378  0.3219833096 
##           230           231           232           233           234 
##  0.0316534704 -0.2581167043  0.0583085955  0.5700003627  0.5954262952 
##           235           236           237           238           239 
##  0.2848515688  0.3462760650 -0.2143090413 -0.5621602103 -0.2730037821 
##           240           241           242           243           244 
## -0.1714468414  0.0391053959  0.2178093368 -0.0687005226 -0.2351469411 
##           245           246           247           248           249 
## -0.1119914021 -0.2920024378 -0.8190123539  0.3833744432  0.0343693485 
##           250           251           252           253           254 
## -0.1799182471  0.1419980348 -0.2378685784  0.0612286255  0.4011171061 
##           255           256           257           258           259 
## -0.2375622209 -0.2020381030 -0.1467294753  0.0970864036 -0.1303181727 
##           260           261           262           263           264 
## -0.2922809287 -0.1274206410  0.0614984843  0.0306651595 -0.0315891801 
##           265           266           267           268           269 
## -0.4377999185 -0.0640159645 -0.0665969726 -0.1057555472 -0.1456231253 
##           270           271           272           273           274 
## -0.2403413712 -0.4356968988  0.2371785611 -0.1979393623 -0.7242058997 
##           275           276           277           278           279 
##  0.3631037673  0.0660423780 -0.3932058015  0.3381009698 -0.1578318148 
##           280           281           282           283           284 
##  0.2446530443 -0.2490493023 -0.0060098678 -0.1666019392 -0.0147796306 
##           285           286           287           288           289 
##  0.1491540493  0.1799059449 -0.2701860074 -0.3336560591 -0.0011273664 
##           290           291           292           293           294 
##  0.1122159231 -1.4486775860 -0.7554797047 -0.5746964907  0.7330382116 
##           295           296           297           298           299 
## -0.3739401710 -0.0293626674 -0.1062926070  0.4555527460 -0.6191352352 
##           300           301           302           303           304 
##  0.2515544686 -0.4484179583 -0.0539196850 -0.0703001396  0.4412054622 
##           305           306           307           308           309 
## -0.0824357825 -0.6025586483 -0.0306639131  0.3063996659  0.0700235724 
##           310           311           312           313           314 
##  0.6314927406 -0.4080936508 -0.1927422244  0.1990798734  0.6016683886 
##           315           316           317           318           319 
##  0.8019766719 -0.3451904574  0.2021213138  0.3879885659  0.3250598275 
##           320           321           322           323           324 
##  0.0189059253 -0.5709861923 -0.1164553442  0.5403781221 -0.2627692245 
##           325           326           327           328           329 
## -0.3169081826 -0.1253828791 -1.2784265672 -0.0530309577  0.6622065640 
##           330           331           332           333           334 
##  0.8513889793  0.5071828163  0.2751323869 -0.1056722935  0.2597791745 
##           335           336           337           338           339 
## -0.1235285176 -0.2939064408  0.2065963386 -0.0239880120 -0.2163514465 
##           340           341           342           343           344 
##  0.5265352521  0.2442196920 -0.8069743153  0.5303510545  0.3661638068 
##           345           346           347           348           349 
##  0.1456767410  0.2254324794 -0.0622707640  0.2238190550  0.3614328646 
##           350           351           352           353           354 
##  0.2400658850  0.3625992808  0.4559347604 -0.1228078173  0.5488429791 
##           355           356           357           358           359 
##  0.3378106674 -0.5142521325 -0.6996584252  0.5752441605  0.8005401107 
##           360           361           362           363           364 
##  0.6971581088  0.5404816367  0.5959684820 -0.1407414080 -0.2717051872 
##           365           366           367           368           369 
##  0.1494480049  0.3511771343  0.2917885557  0.0074824259  0.4390593537 
##           370           371           372           373           374 
##  0.3058360246 -0.0935495226 -0.1221744215  0.1195427603  0.0330591162 
##           375           376           377           378           379 
##  0.0835421921  0.1953184256  0.2587582992 -0.1727205133 -0.2014964087 
##           380           381           382           383           384 
##  0.2696099072  0.4303745555  0.2957669048  0.0586658145 -0.8343168476 
##           385           386           387           388           389 
## -0.6320853484  0.1052786463  0.3578813808 -0.1597636232 -0.0698243549 
##           390           391           392           393           394 
##  0.0748995031  0.0953170765 -0.3862987160 -0.2631336781  0.0133731403 
##           395           396           397           398           399 
## -0.1520176959 -0.0329479126 -0.2016032852 -0.1198533160 -0.4029360466 
##           400           401           402           403           404 
## -0.1847271364 -0.0641867390 -0.0128305703 -0.2279985145 -0.0084782650 
##           405           406           407           408           409 
## -0.0462475260 -0.7000125738  0.0372211934 -0.2341463727 -0.3780974398 
##           410           411           412           413           414 
## -0.0496353806 -0.2560522954 -0.0968365285 -0.4384285480  0.3604897184 
##           415           416           417           418           419 
## -0.0987845652 -0.1786535235  0.0591103709 -0.0073719556 -0.4323526677 
##           420           421           422           423           424 
## -0.5870019956 -0.3829624629 -0.0854455708 -0.8061198733 -0.4404499533 
##           425           426           427           428           429 
##  0.4968442278 -0.3541596113  0.3906029074  0.2237221664  0.2306070112 
##           430           431           432           433           434 
## -0.4696798907  0.2617004917  0.1605134521 -0.3456480710  0.3304342073 
##           435           436           437           438           439 
##  0.0332609729 -0.5598183560  0.0368764989 -0.1381071890 -0.0564951439 
##           440           441           442           443           444 
##  0.1711902095  0.5702508467  0.8419144507 -0.5441600274 -0.0916706866 
##           445           446           447           448           449 
## -0.0946295813 -0.0946213847  0.0025350902  0.1844767897  0.0461788716 
##           450           451           452           453           454 
## -0.3785542075 -0.0668567860 -0.4171291593 -0.0466616887 -0.0843935574 
##           455           456           457           458           459 
##  0.0799719288 -0.0275224466  0.0429623434 -0.2524154955  0.1403735191 
##           460           461           462           463           464 
##  0.7391715299 -0.6646226729 -0.1359374782 -0.1279936971 -0.0095669157 
##           465           466           467           468           469 
##  0.4938193634  0.6910727667 -0.1878455476 -0.1797075425 -0.5611030102 
##           470           471           472           473           474 
## -0.4920585261  0.2236945721  0.1492238259 -0.4326876463 -0.2258603616 
##           475           476           477           478           479 
## -0.5967218108  1.0479218427 -0.7550557877  0.2755654743  0.3655854654 
##           480           481           482           483           484 
##  0.3705433546  0.3725452304  0.6099959922 -0.8424478517 -0.0910911059 
##           485           486           487           488           489 
##  0.2331111264  0.0330803861  0.1782664091 -0.1275199172  0.3380705753 
##           490 
##  0.0008309683

exp(f_ardl914_weather_hanover$forecasts[1,2])
## [1] 1.44021
exp(f_ardl914_weather_hanover$forecasts[1,1])
## [1] 0.6995792
exp(f_ardl914_weather_hanover$forecasts[1,3])
## [1] 3.045355
exp(f_ardl914_weather_hanover$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_hanover_test[1,6])
## [1] 0.332002
exp(f_ardl914_weather_hanover$forecasts[7,2])
## [1] 1.868935
exp(f_ardl914_weather_hanover$forecasts[7,1])
## [1] 0.7130021
exp(f_ardl914_weather_hanover$forecasts[7,3])
## [1] 4.942952
exp(f_ardl914_weather_hanover$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_hanover_test[7,6])
## [1] -0.389472
exp(f_ardl914_weather_hanover$forecasts[14,2])
## [1] 2.388859
exp(f_ardl914_weather_hanover$forecasts[14,1])
## [1] 0.7341548
exp(f_ardl914_weather_hanover$forecasts[14,3])
## [1] 6.731812
exp(f_ardl914_weather_hanover$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_hanover_test[14,6])
## [1] 1.549632
remove <- list(p =list(mean_precipation = c(1:14),
                       mean_temp= c(1:14)))
mod_ardl1414_weather_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                         mean_temp,data = full_cases_wastewater_weather_data_hanover_train, 
                                       p=14,q=14,
                                       remove = remove)
summary(mod_ardl1414_weather_hanover)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.43294 -0.19875  0.00512  0.22626  1.08988 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -0.4670952  0.2786445  -1.676 0.094381 .  
## log_viral_gene.t       0.0922004  0.0301820   3.055 0.002388 ** 
## log_viral_gene.1      -0.0154866  0.0409748  -0.378 0.705645    
## log_viral_gene.2      -0.0526942  0.0409847  -1.286 0.199216    
## log_viral_gene.3       0.0409139  0.0417076   0.981 0.327141    
## log_viral_gene.4       0.0504114  0.0418060   1.206 0.228521    
## log_viral_gene.5      -0.0560534  0.0421717  -1.329 0.184474    
## log_viral_gene.6       0.0043584  0.0423183   0.103 0.918017    
## log_viral_gene.7      -0.0216016  0.0423224  -0.510 0.610021    
## log_viral_gene.8       0.0553156  0.0422959   1.308 0.191610    
## log_viral_gene.9      -0.0876445  0.0423616  -2.069 0.039128 *  
## log_viral_gene.10      0.0087567  0.0420620   0.208 0.835179    
## log_viral_gene.11      0.0611382  0.0420571   1.454 0.146737    
## log_viral_gene.12     -0.0231088  0.0414371  -0.558 0.577341    
## log_viral_gene.13     -0.0195318  0.0414581  -0.471 0.637785    
## log_viral_gene.14     -0.0070912  0.0309912  -0.229 0.819120    
## mean_precipation.t    -0.0692615  0.0422459  -1.639 0.101820    
## mean_temp.t            0.0009596  0.0014032   0.684 0.494412    
## log_mean_new_cases.1   0.4162182  0.0473614   8.788  < 2e-16 ***
## log_mean_new_cases.2   0.1252954  0.0514266   2.436 0.015227 *  
## log_mean_new_cases.3   0.0254412  0.0512488   0.496 0.619840    
## log_mean_new_cases.4   0.1660338  0.0511644   3.245 0.001263 ** 
## log_mean_new_cases.5   0.1277823  0.0516267   2.475 0.013691 *  
## log_mean_new_cases.6   0.1024951  0.0516406   1.985 0.047784 *  
## log_mean_new_cases.7   0.1919035  0.0520010   3.690 0.000252 ***
## log_mean_new_cases.8  -0.0193370  0.0521377  -0.371 0.710901    
## log_mean_new_cases.9  -0.0453096  0.0521161  -0.869 0.385100    
## log_mean_new_cases.10  0.0385100  0.0517716   0.744 0.457364    
## log_mean_new_cases.11 -0.0711252  0.0510386  -1.394 0.164148    
## log_mean_new_cases.12 -0.1561833  0.0510005  -3.062 0.002329 ** 
## log_mean_new_cases.13 -0.0166371  0.0511387  -0.325 0.745082    
## log_mean_new_cases.14  0.0536765  0.0470368   1.141 0.254419    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3718 on 444 degrees of freedom
## Multiple R-squared:  0.9089, Adjusted R-squared:  0.9026 
## F-statistic:   143 on 31 and 444 DF,  p-value: < 2.2e-16
f_ardl1414_weather_hanover <- forecast(mod_ardl1414_weather_hanover, x= t(full_cases_wastewater_weather_data_hanover_test[,c(7,4,5)]),h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
     f_ardl1414_weather_hanover$forecasts) 
## [1] 0.3589144
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
    f_ardl1414_weather_hanover$forecasts) 
## [1] 0.2502069
checkresiduals(mod_ardl1414_weather_hanover)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
##  2.270589e-01 -2.410311e-01 -2.854062e-02  3.283958e-02  1.116750e-01 
##            20            21            22            23            24 
##  2.098113e-01  1.770425e-01  2.020030e-01  3.031835e-02 -1.507969e-02 
##            25            26            27            28            29 
## -2.307308e-02 -2.599690e-01  6.483056e-01 -2.533463e-01  3.131980e-01 
##            30            31            32            33            34 
## -4.168900e-01  1.862153e-01  7.073310e-03 -4.502889e-02  2.929331e-01 
##            35            36            37            38            39 
##  5.636111e-01  2.533170e-01 -1.554401e-01 -1.270906e-01  1.335652e-01 
##            40            41            42            43            44 
##  5.100574e-02  1.784663e-01 -1.903397e-01 -2.833543e-01  1.071710e-01 
##            45            46            47            48            49 
## -1.301165e-01 -1.215357e-01 -2.010920e-01  1.194348e-01 -4.364003e-01 
##            50            51            52            53            54 
##  1.897035e-01 -1.253740e-01 -3.377345e-01  3.549827e-01 -1.616130e-01 
##            55            56            57            58            59 
##  2.351323e-01  4.593852e-01  3.486261e-01  5.279231e-02  4.598862e-02 
##            60            61            62            63            64 
##  1.266101e-01  6.564988e-02  2.885187e-01  2.034467e-02 -2.128808e-01 
##            65            66            67            68            69 
## -3.537216e-01  5.710413e-01  2.231440e-01  1.515378e-01 -1.246995e-01 
##            70            71            72            73            74 
##  4.039996e-01 -6.561615e-02 -7.021196e-02  1.486042e-01 -1.800846e-01 
##            75            76            77            78            79 
##  1.241210e-01  2.429562e-01  1.853579e-01  1.956630e-01  1.019016e-01 
##            80            81            82            83            84 
##  3.981707e-01  2.168091e-01  9.741278e-02 -1.624088e-02 -1.804782e-01 
##            85            86            87            88            89 
##  1.605165e-01 -1.924623e-01  1.704031e-01 -3.871244e-01 -1.060204e-01 
##            90            91            92            93            94 
## -3.125177e-01  1.028900e-01 -2.931317e-01  1.200058e-02 -2.323099e-01 
##            95            96            97            98            99 
##  3.132134e-01  6.486246e-02 -2.600250e-01  2.228510e-01  2.649795e-01 
##           100           101           102           103           104 
##  1.780704e-01  5.836262e-02  4.590045e-02 -3.047084e-02  3.266498e-01 
##           105           106           107           108           109 
## -2.911229e-01 -1.230005e+00  6.487112e-01 -2.004594e-01  1.605844e-01 
##           110           111           112           113           114 
## -4.134065e-02  4.741153e-01  6.270153e-01 -9.873207e-01  7.694942e-01 
##           115           116           117           118           119 
##  2.102079e-01  1.083709e-01 -9.440471e-02 -9.885186e-03 -6.749762e-01 
##           120           121           122           123           124 
##  2.938716e-01  1.033343e-01 -5.714445e-01  5.203362e-02 -1.047539e+00 
##           125           126           127           128           129 
##  2.643079e-01  3.037395e-01 -4.143345e-01  5.311315e-01 -4.646205e-02 
##           130           131           132           133           134 
##  5.165328e-01 -1.668338e-01  1.749692e-01 -4.424815e-01 -1.605759e-01 
##           135           136           137           138           139 
##  1.982563e-01 -1.051565e+00 -4.278540e-01  7.661721e-01  3.480531e-01 
##           140           141           142           143           144 
## -2.606894e-01  1.146758e-01 -5.939083e-01  9.089404e-01 -7.126457e-01 
##           145           146           147           148           149 
##  1.843867e-01 -3.274030e-01  1.535327e-01  9.671142e-02 -4.177192e-01 
##           150           151           152           153           154 
## -1.492836e-01  1.424027e-01 -4.550772e-02 -6.086472e-02  2.530463e-01 
##           155           156           157           158           159 
##  2.403866e-01 -2.852644e-01 -8.697092e-03 -1.486754e-01  2.136880e-02 
##           160           161           162           163           164 
##  5.256361e-02  4.410108e-01  3.082210e-01 -5.653917e-01 -1.594318e-02 
##           165           166           167           168           169 
## -2.104119e-01  4.846740e-02  6.595354e-02  1.794267e-01  1.425456e-01 
##           170           171           172           173           174 
## -2.851416e-01 -1.072190e-01 -2.198649e-02  3.733810e-03 -1.903774e-01 
##           175           176           177           178           179 
##  6.596738e-01 -1.422580e-01 -5.072102e-01 -3.332111e-01 -4.047743e-01 
##           180           181           182           183           184 
##  4.981373e-01 -1.613849e-01 -8.710442e-02  1.975234e-01 -3.284858e-01 
##           185           186           187           188           189 
##  4.366427e-01 -2.929455e-02 -3.150026e-01 -2.887443e-01 -1.110311e-01 
##           190           191           192           193           194 
##  5.828399e-01 -5.054694e-01 -5.554057e-01  1.089883e+00  1.079370e-01 
##           195           196           197           198           199 
##  5.845387e-01 -4.269165e-01 -6.024582e-01 -7.885443e-01  3.826554e-01 
##           200           201           202           203           204 
## -1.090543e-01  7.362514e-01  5.068109e-01  6.526271e-01  7.404637e-01 
##           205           206           207           208           209 
##  2.876024e-01  1.235561e-01 -2.003769e-01 -4.335005e-02 -3.909530e-01 
##           210           211           212           213           214 
## -4.063492e-01  1.302842e-01  9.992110e-02  2.433238e-01 -9.023134e-02 
##           215           216           217           218           219 
##  1.816177e-02  4.684848e-01  3.205087e-02 -3.001481e-02  2.641413e-02 
##           220           221           222           223           224 
##  9.420464e-02 -1.302035e-01 -1.937320e-01  1.056923e-01 -1.769417e-01 
##           225           226           227           228           229 
##  1.650905e-01  1.371480e-01 -6.742795e-02  4.294468e-02  3.394963e-01 
##           230           231           232           233           234 
##  3.623700e-02 -2.380512e-01  8.886582e-02  5.534564e-01  5.494565e-01 
##           235           236           237           238           239 
##  2.701031e-01  3.499940e-01 -1.823197e-01 -5.764624e-01 -3.665840e-01 
##           240           241           242           243           244 
## -2.399389e-01  5.800127e-03  1.456074e-01 -1.744113e-01 -2.503231e-01 
##           245           246           247           248           249 
## -2.593299e-02 -2.109245e-01 -7.470054e-01  5.016789e-01  8.247269e-02 
##           250           251           252           253           254 
## -2.334856e-01  1.352606e-01 -1.413422e-01  1.489468e-01  4.864447e-01 
##           255           256           257           258           259 
## -1.285546e-01 -1.525437e-01 -1.349363e-01 -2.197406e-02 -2.305820e-01 
##           260           261           262           263           264 
## -2.144332e-01 -4.825002e-02  5.905773e-02 -9.467473e-05 -1.012822e-01 
##           265           266           267           268           269 
## -4.390793e-01  1.339791e-02  1.486407e-02 -6.623097e-02 -8.698252e-02 
##           270           271           272           273           274 
## -1.469226e-01 -3.847692e-01  2.592446e-01 -1.563856e-01 -7.027729e-01 
##           275           276           277           278           279 
##  3.872251e-01  1.088777e-01 -4.009366e-01  3.579363e-01 -8.794474e-02 
##           280           281           282           283           284 
##  2.805808e-01 -2.259005e-01 -4.829114e-02 -2.084944e-01  4.464939e-03 
##           285           286           287           288           289 
##  9.735779e-02  8.071994e-02 -2.421299e-01 -2.978718e-01 -3.637500e-02 
##           290           291           292           293           294 
##  1.016082e-01 -1.432937e+00 -7.801099e-01 -5.988665e-01  6.835698e-01 
##           295           296           297           298           299 
## -3.784937e-01  2.075589e-02  6.443711e-02  6.203824e-01 -5.498804e-01 
##           300           301           302           303           304 
##  2.926634e-01 -2.797025e-01 -1.389714e-03 -2.571800e-01  2.357038e-01 
##           305           306           307           308           309 
## -7.824636e-02 -4.796626e-01 -7.408385e-03  2.526975e-01  6.102977e-02 
##           310           311           312           313           314 
##  5.945919e-01 -4.786287e-01 -2.606658e-01  1.495578e-01  5.099051e-01 
##           315           316           317           318           319 
##  7.727305e-01 -2.495181e-01  2.124409e-01  2.689888e-01  2.482887e-01 
##           320           321           322           323           324 
##  1.687323e-02 -4.801578e-01 -6.831247e-02  4.085476e-01 -4.708001e-01 
##           325           326           327           328           329 
## -3.869036e-01 -3.669694e-03 -1.196805e+00 -1.982067e-01  5.794727e-01 
##           330           331           332           333           334 
##  8.667989e-01  5.591952e-01  3.124949e-01 -1.175088e-01  3.100643e-01 
##           335           336           337           338           339 
## -5.628794e-02 -3.417530e-01  1.429134e-01 -1.715225e-01 -5.417116e-01 
##           340           341           342           343           344 
##  3.233918e-01  3.238345e-01 -6.850812e-01  5.301461e-01  2.922436e-01 
##           345           346           347           348           349 
##  5.868424e-02  2.295316e-01 -2.265279e-02  2.586583e-01  4.498998e-01 
##           350           351           352           353           354 
##  2.433672e-01  2.754289e-01  4.642949e-01 -7.955589e-02  4.480241e-01 
##           355           356           357           358           359 
##  3.129450e-01 -5.415047e-01 -7.575037e-01  4.801098e-01  6.654430e-01 
##           360           361           362           363           364 
##  6.247283e-01  5.671282e-01  6.244267e-01 -1.339416e-01 -3.051507e-01 
##           365           366           367           368           369 
##  1.210243e-01  3.805827e-01  2.607641e-01 -2.324668e-01  1.644349e-01 
##           370           371           372           373           374 
##  2.601512e-01  1.885763e-02 -2.036669e-02  1.984214e-01  3.783876e-02 
##           375           376           377           378           379 
## -5.591532e-02  2.700763e-02  2.289714e-01 -8.509009e-02 -1.478307e-01 
##           380           381           382           383           384 
##  2.632013e-01  4.429521e-01  3.046121e-01  2.456644e-02 -8.530184e-01 
##           385           386           387           388           389 
## -6.399881e-01  8.274446e-02  3.372660e-01 -1.147495e-01  5.777909e-03 
##           390           391           392           393           394 
##  8.115420e-02  8.168821e-02 -2.934271e-01 -9.596380e-02  1.711521e-01 
##           395           396           397           398           399 
## -1.072358e-01 -1.756149e-01 -3.297276e-01 -7.229281e-02 -2.626040e-01 
##           400           401           402           403           404 
## -1.192008e-01 -3.874016e-02  4.834980e-02 -1.894929e-01 -2.641341e-02 
##           405           406           407           408           409 
##  2.998860e-03 -5.923718e-01  1.076647e-01 -1.770867e-01 -3.368716e-01 
##           410           411           412           413           414 
## -4.730343e-02 -2.775679e-01 -9.258815e-02 -3.649260e-01  4.165542e-01 
##           415           416           417           418           419 
## -3.994843e-02 -8.101042e-02  1.160832e-01 -1.315062e-02 -3.784094e-01 
##           420           421           422           423           424 
## -5.435940e-01 -4.024176e-01 -1.008230e-01 -7.923297e-01 -4.822691e-01 
##           425           426           427           428           429 
##  4.688669e-01 -2.561113e-01  4.838682e-01  3.077308e-01  3.607276e-01 
##           430           431           432           433           434 
## -3.344516e-01  2.781521e-01  9.937074e-02 -3.407697e-01  3.262542e-01 
##           435           436           437           438           439 
## -8.516648e-02 -6.544093e-01  3.181274e-02 -1.888867e-01 -1.122246e-01 
##           440           441           442           443           444 
##  2.259891e-01  5.806447e-01  7.792650e-01 -5.021219e-01 -6.116585e-02 
##           445           446           447           448           449 
## -1.312755e-01 -9.811521e-02  6.198102e-03  1.064235e-01 -2.131154e-02 
##           450           451           452           453           454 
## -4.170138e-01 -1.298564e-01 -4.328606e-01  6.861778e-02  6.498668e-02 
##           455           456           457           458           459 
##  5.746764e-02 -8.337697e-02  4.740934e-02 -2.839263e-01  1.589511e-01 
##           460           461           462           463           464 
##  8.343013e-01 -5.840918e-01 -1.865652e-01 -1.794954e-01 -7.253902e-02 
##           465           466           467           468           469 
##  3.748417e-01  6.494985e-01 -1.502770e-01 -1.396160e-01 -5.996783e-01 
##           470           471           472           473           474 
## -6.067145e-01  2.147969e-01  1.570543e-01 -5.606905e-01 -3.457215e-01 
##           475           476           477           478           479 
## -6.300701e-01  1.015372e+00 -6.281327e-01  4.363419e-01  4.222513e-01 
##           480           481           482           483           484 
##  3.018794e-01  2.582891e-01  5.830708e-01 -7.478106e-01 -4.883500e-02 
##           485           486           487           488           489 
##  1.577766e-01 -1.902995e-01  2.702728e-02 -1.168220e-01  2.425754e-01 
##           490 
## -9.335678e-02

remove <- list(p =list(mean_precipation = c(1:11),
                       mean_temp= c(1:11)))
mod_ardl1311_weather_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                          mean_temp,data = full_cases_wastewater_weather_data_hanover_train, 
                                        p=11,q=13,
                                        remove = remove)
summary(mod_ardl1311_weather_hanover)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.48048 -0.20447 -0.00048  0.21990  1.07006 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           -0.5562310  0.2710179  -2.052 0.040712 *  
## log_viral_gene.t       0.0903774  0.0301453   2.998 0.002868 ** 
## log_viral_gene.1      -0.0175650  0.0409407  -0.429 0.668104    
## log_viral_gene.2      -0.0497835  0.0408754  -1.218 0.223889    
## log_viral_gene.3       0.0390321  0.0416497   0.937 0.349185    
## log_viral_gene.4       0.0430543  0.0416091   1.035 0.301351    
## log_viral_gene.5      -0.0527710  0.0418437  -1.261 0.207911    
## log_viral_gene.6       0.0083179  0.0420137   0.198 0.843151    
## log_viral_gene.7      -0.0302828  0.0417319  -0.726 0.468431    
## log_viral_gene.8       0.0487226  0.0418839   1.163 0.245334    
## log_viral_gene.9      -0.0754796  0.0411709  -1.833 0.067416 .  
## log_viral_gene.10      0.0118713  0.0412826   0.288 0.773815    
## log_viral_gene.11      0.0213976  0.0308429   0.694 0.488191    
## mean_precipation.t    -0.0708030  0.0421641  -1.679 0.093804 .  
## mean_temp.t            0.0008344  0.0013988   0.597 0.551132    
## log_mean_new_cases.1   0.4170950  0.0471616   8.844  < 2e-16 ***
## log_mean_new_cases.2   0.1142583  0.0507139   2.253 0.024741 *  
## log_mean_new_cases.3   0.0273515  0.0507571   0.539 0.590243    
## log_mean_new_cases.4   0.1693411  0.0507479   3.337 0.000918 ***
## log_mean_new_cases.5   0.1279650  0.0511656   2.501 0.012740 *  
## log_mean_new_cases.6   0.1062070  0.0514707   2.063 0.039644 *  
## log_mean_new_cases.7   0.2079812  0.0510988   4.070 5.55e-05 ***
## log_mean_new_cases.8  -0.0153943  0.0518576  -0.297 0.766713    
## log_mean_new_cases.9  -0.0446710  0.0515833  -0.866 0.386953    
## log_mean_new_cases.10  0.0447135  0.0509052   0.878 0.380214    
## log_mean_new_cases.11 -0.0701170  0.0508681  -1.378 0.168763    
## log_mean_new_cases.12 -0.1571059  0.0502739  -3.125 0.001893 ** 
## log_mean_new_cases.13  0.0009672  0.0463921   0.021 0.983375    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3717 on 449 degrees of freedom
## Multiple R-squared:  0.9082, Adjusted R-squared:  0.9027 
## F-statistic: 164.5 on 27 and 449 DF,  p-value: < 2.2e-16
f_ardl1311_weather_hanover <- forecast(mod_ardl1311_weather_hanover, x= t(full_cases_wastewater_weather_data_hanover_test[,c(7,4,5)]),h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
     f_ardl1311_weather_hanover$forecasts) 
## [1] 0.379521
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
    f_ardl1311_weather_hanover$forecasts)
## [1] 0.2620331
checkresiduals(mod_ardl1311_weather_hanover)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##            14            15            16            17            18 
##  0.2792092891  0.2223521723 -0.2190584141 -0.0004582325  0.0430956794 
##            19            20            21            22            23 
##  0.1140686676  0.2188521455  0.1827547011  0.2120835765  0.0387689363 
##            24            25            26            27            28 
## -0.0153983268 -0.0263591102 -0.2528857831  0.6479000707 -0.2573923315 
##            29            30            31            32            33 
##  0.3278707544 -0.4335560505  0.1767482429 -0.0093173522 -0.0427566444 
##            34            35            36            37            38 
##  0.2964485394  0.5520989204  0.2610007803 -0.1411617878 -0.1548387575 
##            39            40            41            42            43 
##  0.1129916496  0.0359343309  0.2235930161 -0.2176759337 -0.2630381821 
##            44            45            46            47            48 
##  0.1217436895 -0.1125062458 -0.1221446751 -0.2050573223  0.1152355625 
##            49            50            51            52            53 
## -0.4334428939  0.2191539651 -0.0985142772 -0.3269122377  0.3652177348 
##            54            55            56            57            58 
## -0.1591276786  0.2529923775  0.4645336683  0.3418526100  0.0651660934 
##            59            60            61            62            63 
##  0.0498894181  0.1333549535  0.0539981440  0.2568127925 -0.0301109652 
##            64            65            66            67            68 
## -0.2159717327 -0.3619569831  0.5444880064  0.2197994363  0.1223855205 
##            69            70            71            72            73 
## -0.1433320385  0.4050011094 -0.0472916020 -0.1244829019  0.1212334265 
##            74            75            76            77            78 
## -0.1643920577  0.1649665715  0.3148983662  0.2188739493  0.2196154719 
##            79            80            81            82            83 
##  0.1637177077  0.4556781518  0.2328557803  0.1176943491 -0.0523253956 
##            84            85            86            87            88 
## -0.1852388035  0.1561792310 -0.2044685620  0.1575807491 -0.4089961927 
##            89            90            91            92            93 
## -0.1075103202 -0.3493042662  0.0875308187 -0.2827491268  0.0681498800 
##            94            95            96            97            98 
## -0.1838917201  0.3496427178  0.1031812207 -0.2170489164  0.2321295201 
##            99           100           101           102           103 
##  0.2881512421  0.2432880124  0.1087994681  0.0423976575 -0.0275973948 
##           104           105           106           107           108 
##  0.2754592693 -0.3084067285 -1.2425451529  0.6536287786 -0.2254422586 
##           109           110           111           112           113 
##  0.1778788970 -0.0311995188  0.4617557753  0.6247184530 -0.9637168430 
##           114           115           116           117           118 
##  0.7915959699  0.2065183849  0.1334692730 -0.0812679060 -0.0044308145 
##           119           120           121           122           123 
## -0.6729066125  0.2527654281  0.1258199114 -0.5781291148  0.0780429480 
##           124           125           126           127           128 
## -1.0596241181  0.2624293994  0.3399975854 -0.4742550033  0.5573667182 
##           129           130           131           132           133 
## -0.0273250388  0.5439192488 -0.1640590307  0.1986528499 -0.4634757317 
##           134           135           136           137           138 
## -0.1409074048  0.1513350259 -1.0702373820 -0.3815010115  0.7385526952 
##           139           140           141           142           143 
##  0.4135186508 -0.1933050517  0.1186540230 -0.5319396209  0.9428281227 
##           144           145           146           147           148 
## -0.6743453499  0.1741235692 -0.3282807192  0.1465632601  0.0768463947 
##           149           150           151           152           153 
## -0.4370417551 -0.1878569571  0.1104968155 -0.0331266216 -0.0009659063 
##           154           155           156           157           158 
##  0.2642048740  0.2551870499 -0.3571152050  0.0364773733 -0.1678424579 
##           159           160           161           162           163 
##  0.0760519218  0.0734234748  0.4515469937  0.3324520916 -0.5003708848 
##           164           165           166           167           168 
##  0.0121905585 -0.2206893272  0.0309751377  0.0015485862  0.1560330127 
##           169           170           171           172           173 
##  0.1538470293 -0.2965923405 -0.0893191802 -0.0256117117 -0.0049357069 
##           174           175           176           177           178 
## -0.1451797589  0.7007605318 -0.1171548158 -0.5609838151 -0.3297330474 
##           179           180           181           182           183 
## -0.4007532747  0.5156145861 -0.1281270089 -0.0396250894  0.2306807458 
##           184           185           186           187           188 
## -0.3314126297  0.5311022130  0.0110412678 -0.3205536303 -0.3435568100 
##           189           190           191           192           193 
## -0.0967063380  0.5897078940 -0.5231677113 -0.6631972973  1.0295561009 
##           194           195           196           197           198 
##  0.1410488299  0.6216952250 -0.2980279294 -0.5352757961 -0.8046455397 
##           199           200           201           202           203 
##  0.4587783584 -0.1101950535  0.7026397219  0.4671648088  0.5864009355 
##           204           205           206           207           208 
##  0.7635009441  0.2941954226  0.0911131940 -0.1545182065 -0.0589582640 
##           209           210           211           212           213 
## -0.3848578923 -0.3887562572  0.1037023776 -0.0007341046  0.1962100291 
##           214           215           216           217           218 
## -0.1362748238  0.0152615013  0.4562544469  0.0528927392 -0.0050196194 
##           219           220           221           222           223 
##  0.0210566755  0.1077685964 -0.1239874781 -0.1912011574  0.0926152831 
##           224           225           226           227           228 
## -0.1608177914  0.1864151940  0.1286578794 -0.0860102561  0.0332059193 
##           229           230           231           232           233 
##  0.3231851093  0.0447942339 -0.2175589438  0.0973174382  0.5409912839 
##           234           235           236           237           238 
##  0.5362323934  0.2717514349  0.3330050536 -0.1876589389 -0.5636820375 
##           239           240           241           242           243 
## -0.3631886405 -0.2533945272 -0.0599367454  0.0801870131 -0.1930634304 
##           244           245           246           247           248 
## -0.2437629297 -0.0838669835 -0.2578363609 -0.7281421946  0.6096452143 
##           249           250           251           252           253 
##  0.1037700984 -0.2031903407  0.1696213925 -0.2343407797  0.1056715136 
##           254           255           256           257           258 
##  0.5031965342 -0.0242289360 -0.0920323452 -0.1053154291 -0.0284473025 
##           259           260           261           262           263 
## -0.2529086317 -0.2256351353 -0.1077326645  0.0405093192 -0.0100015579 
##           264           265           266           267           268 
## -0.1052045355 -0.4380843364  0.0012626149  0.0055020522 -0.0485837388 
##           269           270           271           272           273 
## -0.0960285994 -0.1432887442 -0.3775066192  0.2956738806 -0.1331706519 
##           274           275           276           277           278 
## -0.6906226520  0.3986158619  0.1412935657 -0.3827672845  0.3736767105 
##           279           280           281           282           283 
## -0.1110182417  0.2574668738 -0.2291004891 -0.0544480392 -0.2394291686 
##           284           285           286           287           288 
## -0.0068378073  0.0639766300  0.0693648943 -0.2512992996 -0.3400284807 
##           289           290           291           292           293 
## -0.0443239024  0.0501794032 -1.4804806186 -0.7742674801 -0.6205856905 
##           294           295           296           297           298 
##  0.6951801328 -0.3911061912  0.0334865252  0.0638419292  0.6343555456 
##           299           300           301           302           303 
## -0.5252855911  0.3356927781 -0.2913467654 -0.0057207774 -0.2456774103 
##           304           305           306           307           308 
##  0.2844134468 -0.1202075309 -0.5039219938 -0.0630668271  0.2525002406 
##           309           310           311           312           313 
##  0.0329964992  0.5758349287 -0.5318091034 -0.2669606511  0.0971245955 
##           314           315           316           317           318 
##  0.4754378591  0.6900214099 -0.2781963391  0.1856142504  0.1733359866 
##           319           320           321           322           323 
##  0.2199022717 -0.0004822604 -0.4634399895 -0.0255919113  0.4190566054 
##           324           325           326           327           328 
## -0.4300051624 -0.2922234057  0.0205397200 -1.2338653080 -0.1987533541 
##           329           330           331           332           333 
##  0.6060958218  0.8299209435  0.5499253179  0.3417917055 -0.1058034160 
##           334           335           336           337           338 
##  0.3285676144 -0.0779490281 -0.3749522976  0.1370796909 -0.1989325265 
##           339           340           341           342           343 
## -0.5708555106  0.3087925196  0.2354777976 -0.7472563495  0.4961084959 
##           344           345           346           347           348 
##  0.2745584850  0.0646891804  0.2418352201 -0.0816197099  0.2474334602 
##           349           350           351           352           353 
##  0.4907970235  0.1433239645  0.2343814142  0.4897963124  0.0919229323 
##           354           355           356           357           358 
##  0.5459981099  0.3396803715 -0.5918853621 -0.7659527793  0.4577546744 
##           359           360           361           362           363 
##  0.6248086019  0.6187538548  0.5305476239  0.6106716818 -0.1302924559 
##           364           365           366           367           368 
## -0.2752522158  0.1098643543  0.3625839019  0.2402798949 -0.2212075333 
##           369           370           371           372           373 
##  0.1713181523  0.2397639598 -0.0049336607 -0.0281550253  0.1936668508 
##           374           375           376           377           378 
##  0.0510488619 -0.0366773338  0.0453516411  0.2124427988 -0.0842425665 
##           379           380           381           382           383 
## -0.1475910204  0.2613032881  0.4406129483  0.2929329946  0.0285230114 
##           384           385           386           387           388 
## -0.8516842219 -0.6466077640  0.0643124972  0.3300621741 -0.1328908878 
##           389           390           391           392           393 
## -0.0040689159  0.0836490455  0.1201035464 -0.2848916010 -0.1120599966 
##           394           395           396           397           398 
##  0.1741114308 -0.0455019076 -0.1282719604 -0.2989890985 -0.0955829816 
##           399           400           401           402           403 
## -0.3229258781 -0.1414844943 -0.0224640605  0.0163903173 -0.2059367988 
##           404           405           406           407           408 
## -0.0108878732  0.0297608960 -0.5966058417  0.1068818363 -0.1668787565 
##           409           410           411           412           413 
## -0.3212534040 -0.0340397134 -0.2767588383 -0.0905960370 -0.4037532477 
##           414           415           416           417           418 
##  0.4004180760 -0.0392553021 -0.1015704616  0.0915091709 -0.0014145000 
##           419           420           421           422           423 
## -0.3422415357 -0.5800023946 -0.4098367138 -0.1008842941 -0.7912262945 
##           424           425           426           427           428 
## -0.4828132137  0.4534928300 -0.2516316402  0.4356903908  0.3157507201 
##           429           430           431           432           433 
##  0.3771111481 -0.3380409860  0.2885982807  0.1135904777 -0.3249176619 
##           434           435           436           437           438 
##  0.2982984274 -0.1160524762 -0.6491538797 -0.0172520094 -0.2464148793 
##           439           440           441           442           443 
## -0.1073706821  0.1893366529  0.5683373929  0.7732573661 -0.4779314841 
##           444           445           446           447           448 
## -0.1648794824 -0.1764759832 -0.0818705451 -0.0069821860  0.1077607792 
##           449           450           451           452           453 
## -0.0074397206 -0.4239312945 -0.1583660965 -0.4271888633  0.0762154092 
##           454           455           456           457           458 
##  0.0585431371  0.1483982242  0.0226254589  0.0501607521 -0.3349673594 
##           459           460           461           462           463 
##  0.1582890856  0.8308830364 -0.5841289706 -0.0853054910 -0.1196018495 
##           464           465           466           467           468 
## -0.0893841679  0.3321793116  0.6276016467 -0.1766019878 -0.1547776862 
##           469           470           471           472           473 
## -0.5477095332 -0.5863557689  0.2084662617  0.0598690552 -0.6016516047 
##           474           475           476           477           478 
## -0.3189491883 -0.6601332739  1.0700598927 -0.6155057022  0.4270998270 
##           479           480           481           482           483 
##  0.3987775740  0.3342462321  0.2639813290  0.5856055878 -0.7829444706 
##           484           485           486           487           488 
## -0.0724945864  0.1459534955 -0.2016262927  0.0100804816 -0.1469747762 
##           489           490 
##  0.1500310128 -0.0589155599

remove <- list(p =list(mean_precipation = c(1:13),
                       mean_temp= c(1:13)))
mod_ardl813_weather_hanover <- ardlDlm(log_mean_new_cases ~ log_viral_gene + mean_precipation +
                                          mean_temp,data = full_cases_wastewater_weather_data_hanover_train, 
                                        p=13,q=8,
                                        remove = remove)
summary(mod_ardl813_weather_hanover)
## 
## Time series regression with "ts" data:
## Start = 14, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.45728 -0.20362 -0.00207  0.23502  1.11040 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          -0.419363   0.280402  -1.496  0.13546    
## log_viral_gene.t      0.089389   0.030632   2.918  0.00370 ** 
## log_viral_gene.1     -0.014273   0.041552  -0.343  0.73139    
## log_viral_gene.2     -0.047902   0.041530  -1.153  0.24934    
## log_viral_gene.3      0.045005   0.042316   1.064  0.28811    
## log_viral_gene.4      0.054980   0.042360   1.298  0.19497    
## log_viral_gene.5     -0.046908   0.042672  -1.099  0.27224    
## log_viral_gene.6     -0.002676   0.042723  -0.063  0.95008    
## log_viral_gene.7     -0.027996   0.042641  -0.657  0.51180    
## log_viral_gene.8      0.068167   0.042583   1.601  0.11012    
## log_viral_gene.9     -0.096333   0.042266  -2.279  0.02312 *  
## log_viral_gene.10     0.007339   0.042473   0.173  0.86289    
## log_viral_gene.11     0.070184   0.041786   1.680  0.09373 .  
## log_viral_gene.12    -0.030315   0.041906  -0.723  0.46981    
## log_viral_gene.13    -0.044443   0.030723  -1.447  0.14871    
## mean_precipation.t   -0.067175   0.042797  -1.570  0.11720    
## mean_temp.t           0.001380   0.001415   0.975  0.33002    
## log_mean_new_cases.1  0.448963   0.046777   9.598  < 2e-16 ***
## log_mean_new_cases.2  0.116268   0.050941   2.282  0.02293 *  
## log_mean_new_cases.3  0.026562   0.051085   0.520  0.60335    
## log_mean_new_cases.4  0.142631   0.050922   2.801  0.00531 ** 
## log_mean_new_cases.5  0.075016   0.050778   1.477  0.14028    
## log_mean_new_cases.6  0.069760   0.050730   1.375  0.16978    
## log_mean_new_cases.7  0.158127   0.050821   3.111  0.00198 ** 
## log_mean_new_cases.8 -0.082524   0.047092  -1.752  0.08038 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3781 on 452 degrees of freedom
## Multiple R-squared:  0.9044, Adjusted R-squared:  0.8993 
## F-statistic: 178.1 on 24 and 452 DF,  p-value: < 2.2e-16
f_ardl813_weather_hanover <- forecast(mod_ardl813_weather_hanover, x= t(full_cases_wastewater_weather_data_hanover_test[,c(7,4,5)]),h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
     f_ardl813_weather_hanover$forecasts) 
## [1] 0.3580285
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases, 
    f_ardl813_weather_hanover$forecasts) 
## [1] 0.2687249
checkresiduals(mod_ardl813_weather_hanover)
## Time Series:
## Start = 14 
## End = 490 
## Frequency = 1 
##           14           15           16           17           18           19 
##  0.201136122  0.138087477 -0.287201701 -0.052076308 -0.002067104  0.080752066 
##           20           21           22           23           24           25 
##  0.196961759  0.203818849  0.252163953 -0.004275758 -0.088601919 -0.041525351 
##           26           27           28           29           30           31 
## -0.279265664  0.673548659 -0.200156773  0.380983246 -0.419303963  0.170739590 
##           32           33           34           35           36           37 
## -0.008480237 -0.074105360  0.262313772  0.590101529  0.230448808 -0.127015624 
##           38           39           40           41           42           43 
## -0.185533122  0.092135946  0.002827951  0.205420878 -0.106952183 -0.232183946 
##           44           45           46           47           48           49 
##  0.105172602 -0.157873823 -0.175422760 -0.296723711  0.062225637 -0.405596962 
##           50           51           52           53           54           55 
##  0.208074750 -0.189459158 -0.422882323  0.250822301 -0.202250881  0.238969847 
##           56           57           58           59           60           61 
##  0.407329096  0.238002899  0.042943606  0.012616767  0.112949498  0.108307049 
##           62           63           64           65           66           67 
##  0.287610278  0.097235278 -0.119365064 -0.292965861  0.646300651  0.233041434 
##           68           69           70           71           72           73 
##  0.108225938 -0.183650869  0.394297110 -0.074574761 -0.081176872  0.102991276 
##           74           75           76           77           78           79 
## -0.228935215  0.085932428  0.289552852  0.223318053  0.160936526 -0.051270758 
##           80           81           82           83           84           85 
##  0.366391729  0.276705565  0.093526577  0.034588297 -0.073395941  0.238345497 
##           86           87           88           89           90           91 
## -0.125824397  0.208149546 -0.406394181 -0.108965029 -0.273728070  0.136231699 
##           92           93           94           95           96           97 
## -0.378166440 -0.099871049 -0.359419933  0.287944665 -0.001975522 -0.329627686 
##           98           99          100          101          102          103 
##  0.138636745  0.216954032  0.140477926  0.065264260  0.031449975 -0.009382325 
##          104          105          106          107          108          109 
##  0.346179115 -0.260776021 -1.194967263  0.658861532 -0.211611811  0.237279301 
##          110          111          112          113          114          115 
## -0.043414688  0.382738622  0.543716848 -1.092734080  0.726163118  0.199963668 
##          116          117          118          119          120          121 
##  0.015315413  0.013144010  0.123717776 -0.644581974  0.264365668  0.038524782 
##          122          123          124          125          126          127 
## -0.495315592 -0.049253232 -1.073232032  0.425858627  0.269940446 -0.580280971 
##          128          129          130          131          132          133 
##  0.525727733 -0.136450551  0.464550899 -0.154115232  0.070464148 -0.514645736 
##          134          135          136          137          138          139 
## -0.214299764  0.256107546 -0.898879542 -0.499129551  0.692321979  0.279988959 
##          140          141          142          143          144          145 
## -0.321280046 -0.002779462 -0.683003447  0.899510819 -0.779246851  0.261569553 
##          146          147          148          149          150          151 
## -0.300546369  0.127442553  0.215723413 -0.296639526 -0.258501183  0.060109899 
##          152          153          154          155          156          157 
## -0.105671274  0.083230131  0.212014478  0.156010293 -0.339707220 -0.073604116 
##          158          159          160          161          162          163 
## -0.203624074 -0.019442747  0.001836799  0.515646723  0.373333552 -0.578922900 
##          164          165          166          167          168          169 
## -0.008737259 -0.181595671 -0.001515983  0.067523034  0.248451690  0.201127717 
##          170          171          172          173          174          175 
## -0.264205286 -0.109144067 -0.022961651 -0.078079936 -0.213062362  0.720606157 
##          176          177          178          179          180          181 
## -0.083290794 -0.494896207 -0.371894574 -0.483660149  0.430160756 -0.211883600 
##          182          183          184          185          186          187 
## -0.045688610  0.276527020 -0.348842787  0.492620205 -0.042111165 -0.409819929 
##          188          189          190          191          192          193 
## -0.251821047  0.010656120  0.712616961 -0.523461543 -0.615014874  1.110403695 
##          194          195          196          197          198          199 
## -0.014583121  0.571812581 -0.442311880 -0.644113868 -0.690624437  0.518256999 
##          200          201          202          203          204          205 
##  0.055437912  0.840896214  0.329558634  0.741237453  0.798165085  0.145797863 
##          206          207          208          209          210          211 
##  0.029371756 -0.134060295  0.100602542 -0.006421255 -0.084638792  0.271700945 
##          212          213          214          215          216          217 
##  0.118366107  0.237147627 -0.070898701 -0.020481936  0.425791827  0.022858482 
##          218          219          220          221          222          223 
##  0.009195603  0.061822830  0.095837710 -0.108671559 -0.178355271  0.093770262 
##          224          225          226          227          228          229 
## -0.168693144  0.185554010  0.195521816 -0.052467983  0.012566510  0.302946097 
##          230          231          232          233          234          235 
##  0.018929675 -0.240448437  0.054451207  0.596966907  0.587624677  0.269479821 
##          236          237          238          239          240          241 
##  0.353936850 -0.189716977 -0.564800048 -0.264751388 -0.129735335  0.064532765 
##          242          243          244          245          246          247 
##  0.206307276 -0.104994852 -0.245976041 -0.158197274 -0.313203433 -0.834779661 
##          248          249          250          251          252          253 
##  0.382023981  0.014486566 -0.169483022  0.109822710 -0.271066626  0.060968286 
##          254          255          256          257          258          259 
##  0.341855818 -0.256220606 -0.143319093 -0.139008526  0.081151818 -0.133540718 
##          260          261          262          263          264          265 
## -0.306678722 -0.121694844  0.074540120  0.013141469 -0.067008563 -0.457237010 
##          266          267          268          269          270          271 
## -0.067574260 -0.100925239 -0.129015371 -0.156833000 -0.256050815 -0.459169970 
##          272          273          274          275          276          277 
##  0.235023648 -0.225913223 -0.718260283  0.364475466  0.048532776 -0.411747868 
##          278          279          280          281          282          283 
##  0.342284523 -0.172462818  0.274749027 -0.264491195 -0.026342688 -0.115555722 
##          284          285          286          287          288          289 
## -0.016207610  0.120630746  0.197674423 -0.283883035 -0.323017563 -0.022510262 
##          290          291          292          293          294          295 
##  0.129532923 -1.457281795 -0.764307068 -0.567108209  0.716424049 -0.435110044 
##          296          297          298          299          300          301 
## -0.058296793 -0.116154673  0.430229801 -0.696427057  0.289145115 -0.419874468 
##          302          303          304          305          306          307 
##  0.011354745 -0.130209319  0.447628080 -0.089265685 -0.580598888 -0.058478793 
##          308          309          310          311          312          313 
##  0.361585196  0.066476608  0.653090805 -0.408456876 -0.147649861  0.180428953 
##          314          315          316          317          318          319 
##  0.575637124  0.836994109 -0.333249395  0.190716004  0.412107404  0.283578329 
##          320          321          322          323          324          325 
## -0.016312212 -0.526851465 -0.065648506  0.546956601 -0.301534385 -0.290340997 
##          326          327          328          329          330          331 
## -0.119437892 -1.219048720 -0.041767727  0.645666082  0.872459278  0.512147959 
##          332          333          334          335          336          337 
##  0.233492431 -0.100141147  0.291078186 -0.145454202 -0.203465597  0.288031194 
##          338          339          340          341          342          343 
## -0.006316665 -0.244808778  0.522306990  0.245303856 -0.772500320  0.530178152 
##          344          345          346          347          348          349 
##  0.368787544  0.145866689  0.194394063 -0.082636688  0.257898695  0.355292668 
##          350          351          352          353          354          355 
##  0.201247279  0.433212991  0.409306787 -0.127008631  0.558126178  0.410382286 
##          356          357          358          359          360          361 
## -0.465281294 -0.676389134  0.599785306  0.804669944  0.704680347  0.528277690 
##          362          363          364          365          366          367 
##  0.617289299 -0.151048840 -0.293539878  0.176683161  0.453347991  0.333282713 
##          368          369          370          371          372          373 
##  0.001341451  0.427572339  0.296529191 -0.130455306 -0.112611804  0.182802995 
##          374          375          376          377          378          379 
##  0.065393493  0.075535696  0.177224380  0.268863200 -0.186241531 -0.213737583 
##          380          381          382          383          384          385 
##  0.293545665  0.442884534  0.285443532  0.052334772 -0.831367730 -0.625694299 
##          386          387          388          389          390          391 
##  0.084921139  0.343263761 -0.149265118 -0.082853122  0.034850680  0.050853476 
##          392          393          394          395          396          397 
## -0.441726859 -0.247528651  0.052062754 -0.143201551 -0.073544362 -0.210568237 
##          398          399          400          401          402          403 
## -0.133618834 -0.419890845 -0.211861999 -0.072464269 -0.010828214 -0.255019683 
##          404          405          406          407          408          409 
## -0.051056485 -0.074760817 -0.716624693  0.019198934 -0.228725922 -0.381313585 
##          410          411          412          413          414          415 
## -0.062253388 -0.281194110 -0.103963640 -0.458847488  0.320646937 -0.096959498 
##          416          417          418          419          420          421 
## -0.203002835  0.043335269 -0.031027742 -0.445895404 -0.584398517 -0.396233642 
##          422          423          424          425          426          427 
## -0.055875259 -0.839551861 -0.459803673  0.489707399 -0.384158699  0.349108584 
##          428          429          430          431          432          433 
##  0.190325628  0.224260317 -0.462528163  0.237546874  0.172775401 -0.302324280 
##          434          435          436          437          438          439 
##  0.312776588  0.065204992 -0.553719692  0.048348811 -0.168060104 -0.027716001 
##          440          441          442          443          444          445 
##  0.172597293  0.546265573  0.858338947 -0.549158098 -0.101947165 -0.064536997 
##          446          447          448          449          450          451 
## -0.116491862  0.012665841  0.202839309  0.062830286 -0.370730330 -0.130823011 
##          452          453          454          455          456          457 
## -0.409409460 -0.055611840 -0.076415875  0.089932454 -0.032477311  0.070029395 
##          458          459          460          461          462          463 
## -0.281919760  0.151533870  0.720220560 -0.651109815 -0.112179683 -0.105471991 
##          464          465          466          467          468          469 
##  0.035219815  0.496528859  0.674641429 -0.169244575 -0.154525846 -0.601122163 
##          470          471          472          473          474          475 
## -0.452664284  0.273196986  0.173809463 -0.416161405 -0.236797656 -0.644270571 
##          476          477          478          479          480          481 
##  1.031363946 -0.779339802  0.332166184  0.395576233  0.361184263  0.351064263 
##          482          483          484          485          486          487 
##  0.605743205 -0.829530446  0.015548050  0.186549694  0.092414220  0.194067125 
##          488          489          490 
## -0.115963642  0.346188494  0.007436453

#plots#

mod_ardl92_acf <- ggAcf(residuals(mod_ardl92)) + theme_bw() 
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -2.396407e-02 -3.976665e-02  2.253188e-02  1.126186e-02 -3.456715e-05 
##            15            16            17            18            19 
## -1.584383e-01 -1.087522e-01 -5.617950e-02 -2.259314e-02 -4.996504e-02 
##            20            21            22            23            24 
##  1.568234e-01  1.735783e-02 -2.082497e-01  2.457965e-01 -5.157538e-02 
##            25            26            27            28            29 
##  2.722065e-03 -2.621960e-01  2.818996e-01 -2.949834e-01 -3.035126e-01 
##            30            31            32            33            34 
##  1.661684e-01 -4.430982e-02  5.693436e-02  1.717116e-01  1.064284e-01 
##            35            36            37            38            39 
## -1.529803e-01 -2.945897e-01  1.263618e-01 -2.567127e-01 -1.473113e-02 
##            40            41            42            43            44 
##  5.214441e-02  2.746484e-02 -2.552454e-01 -8.854845e-02 -1.247769e-01 
##            45            46            47            48            49 
##  1.160506e-01  7.937280e-02 -6.261422e-02  9.856974e-02 -2.195059e-01 
##            50            51            52            53            54 
##  1.127137e-01 -1.766453e-01 -5.426369e-02 -4.002780e-01 -3.406671e-02 
##            55            56            57            58            59 
##  5.963782e-02  1.016042e-01  2.000816e-01  2.983519e-03 -2.650524e-01 
##            60            61            62            63            64 
##  1.486018e-01  1.601307e-01 -1.384878e-01  7.687082e-02 -4.141107e-02 
##            65            66            67            68            69 
##  1.098705e-01 -3.128669e-02  6.173563e-02  1.494617e-01  7.418673e-02 
##            70            71            72            73            74 
##  1.644328e-01  4.680422e-01  2.262063e-02 -1.274640e-01  5.006964e-02 
##            75            76            77            78            79 
## -3.068078e-01  8.782886e-02 -4.406198e-02 -2.869990e-03  5.181883e-03 
##            80            81            82            83            84 
##  5.072444e-02 -2.090606e-01  1.372967e-01 -1.820851e-01 -1.857852e-02 
##            85            86            87            88            89 
##  3.443533e-01 -2.852786e-01  2.303650e-02  2.023233e-01 -2.091241e-01 
##            90            91            92            93            94 
## -5.177255e-01 -1.381966e-01  3.400566e-01  3.701602e-01  2.276316e-01 
##            95            96            97            98            99 
##  1.177870e-01  3.214996e-01  1.667340e-01 -1.459042e-03  2.712643e-01 
##           100           101           102           103           104 
## -3.993105e-01 -1.274003e-02 -2.554181e-01 -1.053282e-01 -1.625378e-01 
##           105           106           107           108           109 
##  5.634557e-02 -2.966006e-02  1.075129e-01 -6.818631e-02  3.926711e-02 
##           110           111           112           113           114 
##  1.489118e-01 -5.768349e-02  1.701619e-01  2.698320e-01  4.012499e-03 
##           115           116           117           118           119 
##  6.400294e-03 -2.979161e-02 -1.944667e-01  5.016915e-02 -2.560521e-01 
##           120           121           122           123           124 
##  3.215355e-01 -2.638250e-01  1.665216e-01 -2.162241e-02  1.748811e-01 
##           125           126           127           128           129 
## -1.494302e-01  1.042969e-02  2.123463e-01 -2.841083e-03  1.362311e-01 
##           130           131           132           133           134 
##  1.325026e-01 -1.897168e-01  1.637814e-01 -3.943942e-02  1.827401e-02 
##           135           136           137           138           139 
## -5.139633e-02  2.554954e-01 -1.630690e-01  1.330048e-01 -2.957152e-01 
##           140           141           142           143           144 
##  1.841674e-01  2.507736e-01 -1.557800e-01  9.174971e-02 -2.204612e-01 
##           145           146           147           148           149 
## -1.119750e-01 -2.616129e-01 -6.771456e-02  1.873420e-01 -1.271189e+00 
##           150           151           152           153           154 
##  2.556409e-01 -1.170900e+00  1.055138e-01  4.032973e-01  3.828688e-01 
##           155           156           157           158           159 
##  5.568852e-01 -2.625216e-01 -3.221344e-01 -3.517223e-01 -2.326498e-01 
##           160           161           162           163           164 
## -6.413548e-01  1.490695e-01 -8.621020e-02  4.664573e-01  1.604656e-01 
##           165           166           167           168           169 
##  1.383717e-01 -2.395036e-02  3.262965e-01 -1.042147e-01 -3.213268e-01 
##           170           171           172           173           174 
## -3.663798e-01 -6.002995e-02 -7.778285e-01  9.547445e-02 -2.161891e-01 
##           175           176           177           178           179 
##  4.416636e-01  6.852906e-02  3.004097e-02 -2.704458e-01 -7.407742e-02 
##           180           181           182           183           184 
##  4.190433e-02  1.797658e-01 -2.287042e-01 -1.620557e+00  7.144359e-01 
##           185           186           187           188           189 
##  2.002879e-01  1.150878e-01  1.289638e-01  6.242632e-01  2.050394e-01 
##           190           191           192           193           194 
##  3.527073e-01 -1.933211e-02  7.994148e-02 -3.369943e-01  1.785146e-01 
##           195           196           197           198           199 
## -2.045247e-01  2.162620e-03  2.513039e-01 -2.552056e-02  3.232862e-01 
##           200           201           202           203           204 
## -6.735815e-02  3.296637e-01  8.652104e-02  2.380897e-01  4.655643e-01 
##           205           206           207           208           209 
##  8.662858e-02 -1.184034e-02 -9.723711e-02  2.729274e-01  8.313439e-03 
##           210           211           212           213           214 
##  1.015981e-01 -5.016132e-02  2.061002e-01 -5.190664e-02  1.400110e-02 
##           215           216           217           218           219 
##  2.209761e-01 -1.501411e-01  5.880174e-02  1.781954e-01  5.708528e-02 
##           220           221           222           223           224 
##  1.981210e-02  1.167935e-02  1.602544e-01  7.165049e-03 -3.680604e-02 
##           225           226           227           228           229 
## -5.092278e-03  1.754901e-01 -3.829182e-02  1.375325e-01  1.482769e-01 
##           230           231           232           233           234 
##  5.776840e-02  1.493228e-01 -9.650623e-02  2.228080e-01 -4.609739e-02 
##           235           236           237           238           239 
##  9.670582e-02  1.313394e-01  1.672115e-01  5.662791e-01 -3.761859e-01 
##           240           241           242           243           244 
## -6.367110e-02 -2.081374e-01  9.186764e-02  2.436832e-01  3.653342e-03 
##           245           246           247           248           249 
## -6.807823e-02 -6.105565e-01 -1.620877e+00  9.680602e-01  1.192657e-03 
##           250           251           252           253           254 
## -9.628339e-02  1.545319e-01  3.129851e-01 -1.631988e-01  2.493072e-01 
##           255           256           257           258           259 
## -4.769542e-01 -8.771902e-02 -2.167812e-03 -1.056218e-02  1.022641e-01 
##           260           261           262           263           264 
## -2.346255e-01  1.379507e-02 -2.023609e-01 -1.173748e-01 -2.047266e-01 
##           265           266           267           268           269 
##  6.049889e-02 -2.346834e-01 -4.229977e-01  3.988999e-01 -9.839163e-02 
##           270           271           272           273           274 
## -2.135352e-02  7.741357e-02  1.877729e-02  2.034479e-01 -4.268264e-01 
##           275           276           277           278           279 
## -1.360064e-01 -2.454923e-01  5.370585e-02 -6.652115e-02  7.476438e-01 
##           280           281           282           283           284 
## -5.191479e-01  4.549196e-01 -2.347329e-01  2.828300e-02  4.456982e-01 
##           285           286           287           288           289 
##  3.968121e-02 -4.771259e-01 -2.831081e-01 -7.737644e-01  3.006624e-01 
##           290           291           292           293           294 
## -1.023992e-01  1.395710e-01 -2.364213e+00 -3.909424e-01  1.450096e+00 
##           295           296           297           298           299 
## -2.085535e-01  2.979523e-01  6.068575e-01  3.951824e-01  2.037335e-01 
##           300           301           302           303           304 
##  8.261871e-02 -3.236322e-01 -5.990560e-01 -2.930634e-01 -2.168600e-01 
##           305           306           307           308           309 
## -4.583283e-02 -3.267313e-01  1.914202e-01 -1.029380e-01  4.824966e-01 
##           310           311           312           313           314 
## -2.567830e-01 -9.254083e-01  5.238682e-01 -2.883445e-01  4.218525e-01 
##           315           316           317           318           319 
## -1.152390e-01  2.896990e-01  2.318420e-01 -4.225929e-02  1.206934e-01 
##           320           321           322           323           324 
## -1.258189e-02  8.128629e-02  5.362193e-01  3.580403e-02  5.387543e-02 
##           325           326           327           328           329 
## -1.426912e-01 -2.952867e-01 -7.912501e-01  1.631587e-01  4.219926e-01 
##           330           331           332           333           334 
##  6.565432e-02  4.054452e-01  2.009018e-01  4.716507e-01  2.343397e-02 
##           335           336           337           338           339 
##  2.490811e-01  5.407757e-02  2.453663e-01 -2.296027e-01  3.777055e-02 
##           340           341           342           343           344 
## -2.720543e-01 -4.027320e-01  2.307273e-01 -3.164816e-02  2.747526e-01 
##           345           346           347           348           349 
## -4.497552e-03 -1.574489e-01 -4.906954e-02  8.772337e-02  1.370770e-01 
##           350           351           352           353           354 
##  4.529530e-01  1.906016e-01  1.525708e-01  3.284963e-01  3.650998e-01 
##           355           356           357           358           359 
##  5.396324e-01 -3.074325e-01 -5.349434e-01  8.777813e-01  9.443478e-01 
##           360           361           362           363           364 
##  7.003203e-01  4.952868e-01  5.047689e-01 -1.918388e-01  5.261175e-02 
##           365           366           367           368           369 
## -1.077546e-01  1.598769e-01  4.668144e-01  3.266047e-01  2.771328e-01 
##           370           371           372           373           374 
##  1.589788e-01  1.313453e-02 -6.308058e-01  3.184335e-01 -6.444854e-03 
##           375           376           377           378           379 
##  1.939464e-01  1.715339e-01  1.996648e-01 -3.091449e-01 -1.296596e+00 
##           380           381           382           383           384 
##  1.088063e-01  5.960841e-01  2.713044e-01  2.329620e-01 -8.556874e-01 
##           385           386           387           388           389 
## -9.022326e-01  4.009147e-01  5.146857e-01 -9.337388e-02  1.724957e-01 
##           390           391           392           393           394 
##  1.810539e-01  1.784066e-01 -8.606756e-01 -2.221742e-01  1.713428e-01 
##           395           396           397           398           399 
## -1.589677e-01 -2.242434e-03  6.841561e-02  4.039676e-02  1.309074e-01 
##           400           401           402           403           404 
## -1.937431e-01 -1.420454e-01 -2.128705e-01  1.629227e-02 -2.543536e-01 
##           405           406           407           408           409 
## -2.542008e-01  5.524668e-03  3.713062e-02 -3.802828e-02 -8.559204e-02 
##           410           411           412           413           414 
## -1.088547e-01  9.989792e-02 -1.654432e-01 -3.312310e-01  2.208590e-01 
##           415           416           417           418           419 
## -1.385087e-01 -2.478310e-01 -2.713380e-01 -1.457082e-01 -6.735750e-02 
##           420           421           422           423           424 
## -5.499413e-01  1.577297e-02 -2.629274e-01 -2.215989e-01  1.088316e-01 
##           425           426           427           428           429 
##  1.518798e-01 -7.800507e-01  4.120184e-02  8.104201e-02  1.116859e-01 
##           430           431           432           433           434 
##  6.589971e-02 -5.915863e-01  3.969890e-01 -1.167412e-01  5.200500e-02 
##           435           436           437           438           439 
## -1.569589e-01  6.584974e-02 -5.131337e-01  1.382272e-01 -1.724006e-01 
##           440           441           442           443           444 
## -9.139437e-02 -9.756700e-02  4.205117e-01  9.049092e-02 -4.838474e-01 
##           445           446           447           448           449 
##  7.527956e-03  3.232683e-01 -1.284454e-01 -3.784562e-01 -9.697398e-02 
##           450           451           452           453           454 
## -3.759760e-01  3.639162e-01 -2.447045e-01 -3.919063e-02  5.256440e-01 
##           455           456           457           458           459 
## -1.279611e-01  2.734360e-01  6.062632e-02 -2.399165e-01  3.003080e-01 
##           460           461           462           463           464 
## -4.265930e-02  5.641235e-02  3.142238e-01  2.458586e-02 -1.890712e-01 
##           465           466           467           468           469 
##  2.477738e-01  2.543826e-01  8.183976e-02 -1.401468e-01  3.270317e-01 
##           470           471           472           473           474 
## -8.675708e-02 -2.404920e-02  1.507883e-01 -1.698617e-01 -3.771766e-02 
##           475           476           477           478           479 
##  9.785553e-02  7.054171e-02  3.302712e-01 -2.662845e-01  8.830143e-03 
##           480           481           482           483           484 
##  2.716887e-03 -1.649914e-01  1.160813e-01  1.064102e-01 -1.588147e-01 
##           485           486           487           488           489 
##  1.602108e-01  3.967555e-02  3.183249e-02  2.427844e-01 -4.431468e-03 
##           490 
##  2.543166e-01
mod_ardl92_meck_acf <- ggAcf(residuals(mod_ardl92_meck)) + theme_bw()
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -3.092371e-02  1.163238e-02  1.252001e-02  1.014782e-01  2.836953e-02 
##            15            16            17            18            19 
##  2.833400e-02 -1.696518e-03 -1.227269e-01 -1.590525e-01 -3.045078e-02 
##            20            21            22            23            24 
##  2.184542e-01  5.078416e-02 -3.255922e-02 -1.214547e-01  2.311235e-01 
##            25            26            27            28            29 
##  4.023654e-02 -2.353277e-01 -8.977339e-02  1.961115e-01  5.112010e-03 
##            30            31            32            33            34 
## -3.149867e-02  1.427563e-02 -2.207462e-01 -2.851758e-02 -2.929289e-01 
##            35            36            37            38            39 
##  3.177582e-01  4.487638e-02 -9.499900e-02 -8.387231e-02 -2.265090e-01 
##            40            41            42            43            44 
##  8.259352e-02  9.959311e-02 -8.641751e-02 -1.764461e-01  1.229113e-01 
##            45            46            47            48            49 
## -7.265117e-02 -2.519822e-01  9.349147e-02 -2.707908e-01  3.195785e-01 
##            50            51            52            53            54 
## -2.050264e-01 -3.964405e-01  1.363230e-01  1.925282e-02  2.139667e-02 
##            55            56            57            58            59 
## -1.014455e-01 -1.146824e-01  3.385559e-01 -3.207601e-01  1.507240e-01 
##            60            61            62            63            64 
## -1.071141e+00 -2.682274e-01  2.178915e-01 -5.722560e-01  6.045407e-01 
##            65            66            67            68            69 
##  3.489794e-01  2.040119e-01 -3.519632e-02  2.839743e-01  2.117074e-01 
##            70            71            72            73            74 
##  3.177930e-01 -4.062772e-02 -6.585365e-02 -2.198761e-01  4.218497e-01 
##            75            76            77            78            79 
## -1.655438e-01  4.272526e-01  3.326260e-01  3.096256e-01 -1.568450e-02 
##            80            81            82            83            84 
## -6.525427e-02 -1.434785e-01 -3.334671e-01 -2.611591e-02 -4.666716e-02 
##            85            86            87            88            89 
##  3.474433e-01 -2.150304e-01  3.881112e-01  2.499551e-02  2.028181e-01 
##            90            91            92            93            94 
## -2.367724e-01 -8.491285e-02  1.206571e-02 -3.959261e-01  3.882358e-02 
##            95            96            97            98            99 
## -2.155568e-01  1.809715e-01  8.645893e-02  2.824313e-03 -1.786669e-02 
##           100           101           102           103           104 
## -8.591058e-02  2.274223e-01 -1.068923e-01 -3.708266e-01  3.465886e-01 
##           105           106           107           108           109 
## -4.702015e-01  6.073526e-01 -6.934208e-02 -1.380305e-01  1.090499e-01 
##           110           111           112           113           114 
## -3.081340e-01  9.294370e-02  2.802856e-01 -1.150670e-01 -4.287394e-02 
##           115           116           117           118           119 
## -1.069286e-01 -4.091438e-01  3.187158e-01 -3.036462e-01 -8.434482e-02 
##           120           121           122           123           124 
## -1.245349e-01  4.561678e-02 -1.409853e-01 -2.357035e-02 -3.462645e-01 
##           125           126           127           128           129 
## -4.734365e-03  3.363099e-01 -2.809807e-01 -6.658337e-01 -1.931057e-01 
##           130           131           132           133           134 
## -8.923531e-01  9.462148e-02 -1.521770e-01  7.699534e-01  7.831224e-02 
##           135           136           137           138           139 
##  1.447998e-01 -2.591512e-01 -2.804563e-01  1.452160e-01 -5.340732e-01 
##           140           141           142           143           144 
## -1.774752e-01  3.831458e-01 -2.355669e-01 -5.379080e-01  4.430436e-01 
##           145           146           147           148           149 
##  4.268551e-01  7.313508e-02  8.329347e-02 -2.276817e-01 -3.157568e-01 
##           150           151           152           153           154 
##  9.449980e-01 -1.127343e-02 -7.474956e-01 -3.350319e-01  6.629206e-01 
##           155           156           157           158           159 
## -3.926999e-01 -2.283320e-01 -2.171595e-01  3.702117e-01  2.552329e-01 
##           160           161           162           163           164 
## -3.746085e-01  3.234608e-01  3.548603e-01  3.353210e-01  1.897544e-01 
##           165           166           167           168           169 
## -3.546162e-01 -2.689742e-01 -1.656706e-01  3.926646e-01 -5.638034e-01 
##           170           171           172           173           174 
## -6.236650e-01 -3.644881e-01  2.419904e-01  6.867030e-02 -7.785484e-01 
##           175           176           177           178           179 
##  3.596520e-01  2.556221e-01 -2.183095e-01  5.990763e-02 -4.009686e-01 
##           180           181           182           183           184 
##  9.059714e-01  2.544917e-01  1.393752e-01 -4.475876e-01  5.457393e-02 
##           185           186           187           188           189 
## -2.374567e-01  8.340786e-02  4.492369e-01  4.403853e-01  5.910953e-01 
##           190           191           192           193           194 
##  2.577866e-01 -1.954837e-01  1.473030e-01 -1.985440e-01 -4.717432e-02 
##           195           196           197           198           199 
##  7.214516e-02 -1.357494e-02  7.170069e-01 -1.940442e-02  1.830457e-01 
##           200           201           202           203           204 
##  1.607166e-01  1.394537e-01  2.803610e-01  1.726479e-01  1.823049e-01 
##           205           206           207           208           209 
##  3.561684e-02 -9.836023e-02  1.927115e-01  6.261834e-02 -1.391065e-02 
##           210           211           212           213           214 
##  1.735067e-01  1.859842e-01  7.134149e-02 -1.744462e-01 -5.452612e-02 
##           215           216           217           218           219 
##  1.558357e-01 -3.467078e-02 -1.510130e-01  1.601537e-01 -2.934839e-02 
##           220           221           222           223           224 
##  5.364477e-02  1.838195e-01  3.012772e-02  9.895059e-02 -2.495102e-01 
##           225           226           227           228           229 
## -1.737132e-02 -1.475369e-02  1.207811e-01  8.583218e-02  1.213871e-01 
##           230           231           232           233           234 
##  1.855354e-01  4.753981e-02  1.866039e-01  1.354519e-01 -1.621541e-01 
##           235           236           237           238           239 
## -1.476087e-01 -9.523335e-03  1.469191e-01 -5.875478e-02 -6.073574e-02 
##           240           241           242           243           244 
##  1.394707e-01 -9.513615e-02 -2.445320e-01  8.757945e-02 -2.934725e-02 
##           245           246           247           248           249 
## -9.233432e-02  2.892464e-02 -8.366786e-01  5.078695e-01  2.107009e-01 
##           250           251           252           253           254 
##  2.047745e-01  8.030800e-02  1.051612e-01 -1.453955e-01  9.064731e-02 
##           255           256           257           258           259 
## -1.313186e-01 -4.096315e-01 -1.006922e-02  7.446587e-02 -1.236577e-01 
##           260           261           262           263           264 
## -1.928842e-01  6.074715e-02 -1.290528e-01 -2.034470e-01  7.606261e-02 
##           265           266           267           268           269 
## -1.389784e-01 -2.766717e-02 -2.661200e-01  1.163460e-01 -2.396346e-01 
##           270           271           272           273           274 
##  4.097262e-01 -4.466364e-02  1.329093e-01 -1.376155e-01 -3.050477e-01 
##           275           276           277           278           279 
##  6.348342e-02 -1.009835e-01 -3.900220e-01 -1.174700e-01  1.093365e-01 
##           280           281           282           283           284 
## -2.734615e-01  4.514930e-02 -1.382426e-02 -1.349219e-01  1.037062e-01 
##           285           286           287           288           289 
## -1.236246e-01  1.722392e-01  6.008186e-02 -1.184771e-01 -1.498106e-01 
##           290           291           292           293           294 
## -3.679851e-01 -1.146621e-01 -1.211610e+00 -2.829368e-01  6.819258e-01 
##           295           296           297           298           299 
##  5.833887e-01  5.790571e-02  1.464892e-01  3.321741e-01 -3.720783e-01 
##           300           301           302           303           304 
##  1.564222e-01  1.168640e-01 -2.581056e-01  1.968431e-02  2.939600e-02 
##           305           306           307           308           309 
## -5.594626e-01  2.314759e-01 -7.625572e-02  1.770047e-01  6.695224e-02 
##           310           311           312           313           314 
##  1.017863e-01 -2.388097e-01  2.251281e-03 -3.007638e-02  6.243413e-02 
##           315           316           317           318           319 
##  1.010172e-01  2.798638e-01 -1.123650e-01  1.994614e-01  2.516946e-02 
##           320           321           322           323           324 
##  1.304755e-02 -9.812070e-02  1.101769e-01 -5.688409e-02 -1.611528e-01 
##           325           326           327           328           329 
##  2.100385e-01 -4.400001e-02 -5.062325e-01  2.867265e-01  8.064437e-01 
##           330           331           332           333           334 
##  3.036679e-02  3.228944e-01 -6.375375e-02  2.207628e-01  3.847121e-02 
##           335           336           337           338           339 
##  4.157262e-01 -1.862650e-01 -2.152798e-01  1.450628e-01 -1.573700e-01 
##           340           341           342           343           344 
## -3.183065e-02 -9.206769e-02  1.181346e-01  7.614366e-02  2.500905e-02 
##           345           346           347           348           349 
##  3.277367e-01  7.651524e-03  2.772196e-01  2.234536e-01  3.496113e-01 
##           350           351           352           353           354 
##  1.703349e-01  2.441747e-01  2.443558e-01  3.307703e-01  4.035216e-01 
##           355           356           357           358           359 
##  3.452711e-01 -2.891394e-02 -7.149718e-01  7.580044e-01  6.009496e-01 
##           360           361           362           363           364 
##  5.801873e-01  4.424346e-01  4.607344e-01 -7.361956e-02 -2.887587e-01 
##           365           366           367           368           369 
##  2.649712e-01  2.405537e-01  2.796954e-01  2.958525e-01  2.132516e-01 
##           370           371           372           373           374 
## -2.750058e-02 -2.824429e-01 -1.764350e-01  2.016469e-01 -1.351666e-02 
##           375           376           377           378           379 
##  1.020631e-01  6.526443e-02  1.155335e-01 -2.553103e-01 -1.762980e+00 
##           380           381           382           383           384 
##  2.101670e-01  5.359992e-01  4.782681e-01  3.562092e-01 -8.033185e-02 
##           385           386           387           388           389 
## -3.883561e-01  2.627346e-02  7.952609e-02 -2.607643e-01 -1.175277e-01 
##           390           391           392           393           394 
## -5.857509e-02 -1.150789e-01 -2.598925e-01 -4.213645e-02  1.198930e-01 
##           395           396           397           398           399 
## -2.661433e-01  1.489386e-01 -2.024840e-01  7.704819e-02 -2.903232e-01 
##           400           401           402           403           404 
## -1.647626e-01  1.643158e-01 -1.258556e-01  7.388809e-02 -7.547620e-02 
##           405           406           407           408           409 
##  4.738201e-02 -1.695564e-01 -5.873658e-01  1.816129e-01 -3.658345e-01 
##           410           411           412           413           414 
## -1.541965e-01 -1.419844e-01  8.592332e-03 -7.089878e-03  1.165630e-01 
##           415           416           417           418           419 
## -6.272235e-02 -5.778883e-02 -2.247626e-01 -2.778469e-01  4.572784e-01 
##           420           421           422           423           424 
## -6.440206e-01 -1.030302e+00  3.249665e-01 -4.085808e-01 -1.785736e-01 
##           425           426           427           428           429 
##  7.035872e-02 -3.651298e-01 -5.662069e-01  7.933071e-01 -4.768814e-01 
##           430           431           432           433           434 
## -2.922878e-01 -2.685913e-01 -7.243048e-01  1.725235e-01  1.543828e-01 
##           435           436           437           438           439 
## -1.620727e-01 -4.584507e-01  5.191641e-02  9.345671e-01 -3.536851e-01 
##           440           441           442           443           444 
##  6.352328e-02 -3.436279e-01  1.305824e-01  2.842651e-01 -3.008418e-01 
##           445           446           447           448           449 
##  1.386451e-01  1.215209e-01 -2.017925e-01 -4.699615e-01  4.751846e-01 
##           450           451           452           453           454 
## -7.258056e-01  1.091502e+00  1.442230e-01 -1.525797e-02 -4.045037e-01 
##           455           456           457           458           459 
##  1.532905e-02 -3.790277e-02  8.403449e-02 -3.366523e-02  2.058621e-01 
##           460           461           462           463           464 
##  2.752573e-01 -1.232036e-01 -5.652366e-01  4.062207e-01  4.750853e-01 
##           465           466           467           468           469 
##  1.755680e-01  3.300130e-01  3.608029e-01 -3.666815e-01 -5.185949e-02 
##           470           471           472           473           474 
## -6.779433e-01  3.216355e-01  7.937263e-02  5.339369e-02  1.819193e-01 
##           475           476           477           478           479 
## -1.874273e-01 -3.059841e-01  8.937394e-03  4.802756e-01 -1.195863e-01 
##           480           481           482           483           484 
##  2.334788e-01 -2.064050e-02 -4.314131e-02  6.413132e-02 -4.859655e-03 
##           485           486           487           488           489 
##  1.579740e-01 -5.826499e-02  1.890077e-01  6.867093e-02 -4.940313e-05 
##           490 
##  1.385353e-01
mod_ardl92_hanover_acf <- ggAcf(residuals(mod_ardl92_hanover)) + theme_bw()
## Time Series:
## Start = 10 
## End = 490 
## Frequency = 1 
##            10            11            12            13            14 
## -0.2937247715  0.0725525051  0.2357132291  0.0428823729  0.2024050180 
##            15            16            17            18            19 
##  0.1383610625 -0.2676940703 -0.0134268985 -0.0117135696  0.0764144973 
##            20            21            22            23            24 
##  0.1981303419  0.2113455282  0.2379975457  0.0051074184 -0.0491099861 
##            25            26            27            28            29 
## -0.1261627848 -0.2905892448  0.6591943825 -0.2276926085  0.3194060555 
##            30            31            32            33            34 
## -0.4551185306  0.1445597583 -0.0568366994 -0.0627825150  0.2938045965 
##            35            36            37            38            39 
##  0.3709020208  0.2960118006 -0.0942220382 -0.1734517743  0.0226241743 
##            40            41            42            43            44 
##  0.0270041027  0.1869636936 -0.2392370427 -0.3142958568  0.2411378355 
##            45            46            47            48            49 
## -0.0771738354 -0.1404487333 -0.3677277543  0.0431663383 -0.4316978812 
##            50            51            52            53            54 
##  0.2531292334 -0.1388129807 -0.4180790801  0.3174781199 -0.1916208740 
##            55            56            57            58            59 
##  0.1987007527  0.3663651337  0.2555273155  0.0239482523 -0.0530943676 
##            60            61            62            63            64 
##  0.1176959277  0.0485663629  0.1937034479  0.0685978034 -0.1774245281 
##            65            66            67            68            69 
## -0.4345144209  0.6653127842  0.1917715091  0.1800176281 -0.2190088185 
##            70            71            72            73            74 
##  0.4411357528  0.0434507919 -0.0425528122  0.0906556734 -0.1789678336 
##            75            76            77            78            79 
##  0.1817007557  0.4857756943  0.2444219798  0.1366642375  0.1156306111 
##            80            81            82            83            84 
##  0.4289515301  0.3184282752  0.1221898253 -0.0488892873 -0.1376011723 
##            85            86            87            88            89 
##  0.2397033861 -0.0037757774  0.2027015932 -0.3899420104 -0.0166538988 
##            90            91            92            93            94 
## -0.2101998536  0.0999245294 -0.2764622533  0.1196021546 -0.1969040255 
##            95            96            97            98            99 
##  0.4246182427  0.0866703371 -0.2692966292  0.1022241170  0.1845049143 
##           100           101           102           103           104 
##  0.2844814622  0.0567238754  0.0718908996 -0.0193711583  0.3085424980 
##           105           106           107           108           109 
## -0.2717931726 -1.1953301375  0.6730721267 -0.1633869109  0.3155728050 
##           110           111           112           113           114 
## -0.0091031336  0.3984023024  0.5548959808 -1.0566851701  0.7880259146 
##           115           116           117           118           119 
##  0.0939056670  0.1223389249  0.0491446447  0.1406920939 -0.6743811810 
##           120           121           122           123           124 
##  0.3197780047  0.2157982532 -0.5256451527  0.0489862102 -1.0499510287 
##           125           126           127           128           129 
##  0.4394807412  0.3652649330 -0.5195959156  0.3998504377 -0.1191334670 
##           130           131           132           133           134 
##  0.4433747581 -0.0947424623  0.0832120425 -0.5277903645 -0.1063871747 
##           135           136           137           138           139 
##  0.2841217598 -0.8616391162 -0.3790968311  0.7288172894  0.5071672677 
##           140           141           142           143           144 
## -0.2286002821  0.0314926421 -0.6829332184  0.9703729179 -0.7449079747 
##           145           146           147           148           149 
##  0.2679769867 -0.3756868031  0.2328898204  0.2944174911 -0.4603615280 
##           150           151           152           153           154 
## -0.2255009541 -0.0717132127 -0.1898146588  0.0234792637  0.3163682613 
##           155           156           157           158           159 
##  0.2111669824 -0.2559877586 -0.0066921052 -0.1121490904 -0.0041986795 
##           160           161           162           163           164 
##  0.0424280537  0.3866164312  0.3249911181 -0.4301015712 -0.0217016916 
##           165           166           167           168           169 
## -0.2131773712  0.0505797912  0.0663116210  0.2714930485  0.2043053345 
##           170           171           172           173           174 
## -0.3305856683  0.0199390521 -0.0987236863 -0.0879896828 -0.1770255319 
##           175           176           177           178           179 
##  0.9075754923 -0.1057388530 -0.4752423733 -0.1321610850 -0.3801484525 
##           180           181           182           183           184 
##  0.5419094734 -0.2656456412  0.1227014611  0.2842934188 -0.3350889685 
##           185           186           187           188           189 
##  0.5033548382 -0.0068548374 -0.6091154625 -0.2503721300  0.0775616818 
##           190           191           192           193           194 
##  0.8245964949 -0.3712032240 -0.6583240983  1.1415074891  0.0754030017 
##           195           196           197           198           199 
##  0.6061736284 -0.2877254417 -0.7529277730 -0.8836332442  0.7384886742 
##           200           201           202           203           204 
##  0.0897824100  0.7422410308  0.3995787787  0.7494566948  0.8276606603 
##           205           206           207           208           209 
##  0.1200761219 -0.0654303538 -0.2230863545  0.0274122666 -0.0400149964 
##           210           211           212           213           214 
##  0.0022512118  0.2936783139  0.0990719213  0.0543947860 -0.0493149906 
##           215           216           217           218           219 
## -0.0373004111  0.3833824395  0.0185383025  0.0589516846  0.0781896085 
##           220           221           222           223           224 
##  0.1379129373 -0.0118587807 -0.1957035215  0.1029201034 -0.1096721278 
##           225           226           227           228           229 
##  0.2728494624  0.2057127592 -0.1336138055  0.0636875549  0.2787792813 
##           230           231           232           233           234 
## -0.0819055576 -0.2028231242  0.1279166587  0.6104019865  0.5059773785 
##           235           236           237           238           239 
##  0.2927907643  0.3309247297 -0.2103299644 -0.6020374951 -0.2870039810 
##           240           241           242           243           244 
## -0.1242200552  0.1387655073  0.1208341292  0.1154879578 -0.1883789764 
##           245           246           247           248           249 
## -0.1700820686 -0.4658964526 -0.8623299961  0.6775253499  0.0105449122 
##           250           251           252           253           254 
## -0.0087829510  0.1708893966 -0.2762265336 -0.0240055186  0.3459538609 
##           255           256           257           258           259 
## -0.1418843551 -0.1564006393 -0.1053706580  0.1066407105 -0.1853499335 
##           260           261           262           263           264 
## -0.2487433641 -0.2333642650 -0.2727303989 -0.1687733447 -0.0466078025 
##           265           266           267           268           269 
## -0.3995324342 -0.0677488924 -0.0471187115 -0.0601087398 -0.1041610652 
##           270           271           272           273           274 
## -0.2367197351 -0.3989378404  0.2484457684 -0.1568777389 -0.7453024532 
##           275           276           277           278           279 
##  0.3555214858  0.1002671666 -0.3868654278  0.3040895309 -0.2028461658 
##           280           281           282           283           284 
##  0.1722189512 -0.2332142707 -0.0440658053 -0.3086458206 -0.0504900649 
##           285           286           287           288           289 
##  0.1012008512  0.1700433893 -0.3481999153 -0.3257685419  0.0074100728 
##           290           291           292           293           294 
##  0.0341751172 -1.4888343080 -0.7335710249 -0.5480315814  0.7716219916 
##           295           296           297           298           299 
## -0.3682446885 -0.0486605721 -0.1022491050  0.4153375971 -0.6283894824 
##           300           301           302           303           304 
##  0.2012134868 -0.5205160486 -0.1192180476 -0.0939335491  0.4307161292 
##           305           306           307           308           309 
## -0.1346345308 -0.6418514036 -0.1222579844  0.1329938855  0.1042333559 
##           310           311           312           313           314 
##  0.5206664141 -0.6846115233 -0.3201812611  0.0842644910  0.6126850050 
##           315           316           317           318           319 
##  0.5875763431 -0.4078496593  0.2353657459  0.3752687472  0.3007912347 
##           320           321           322           323           324 
##  0.0219233387 -0.5585097529  0.0920338782  0.4985910974 -0.2534512110 
##           325           326           327           328           329 
## -0.1352546575 -0.0338617618 -1.2401356127 -0.0603990799  0.6927204590 
##           330           331           332           333           334 
##  0.8101645216  0.4554625066  0.2758977507 -0.1558902425  0.2324918101 
##           335           336           337           338           339 
## -0.1571132546 -0.3554635846  0.1720243647 -0.0368614666 -0.2860883750 
##           340           341           342           343           344 
##  0.3664146703  0.3245726439 -0.8113220120  0.3094211668  0.1261280314 
##           345           346           347           348           349 
##  0.1013780624  0.5355813115 -0.2189059879  0.4203520863  0.4150589674 
##           350           351           352           353           354 
##  0.3031770151  0.1136186294  0.3525790847  0.0603945804  0.6853335007 
##           355           356           357           358           359 
##  0.3605603211 -0.5404365110 -0.6752436627  0.5970516429  0.8297212770 
##           360           361           362           363           364 
##  0.7278450588  0.5652834562  0.5630755331 -0.1175717912 -0.1903461945 
##           365           366           367           368           369 
##  0.0972404074  0.3422963065  0.3265283580  0.0821512567  0.4221394520 
##           370           371           372           373           374 
##  0.3144995798  0.0242426290 -0.0869549654  0.1292901847  0.0371106374 
##           375           376           377           378           379 
##  0.1510399797  0.1887376894  0.2498141128 -0.1476159290 -0.2584843004 
##           380           381           382           383           384 
##  0.2742931587  0.3827214873  0.2861100347  0.0320705553 -0.8687612922 
##           385           386           387           388           389 
## -0.6717371112  0.0901594979  0.3829023442 -0.0934348061 -0.0938720504 
##           390           391           392           393           394 
##  0.1243421596  0.0823670432 -0.4012690476 -0.3010281636 -0.0453070090 
##           395           396           397           398           399 
## -0.1682664417 -0.0227266962 -0.1872004916 -0.1452224339 -0.5144250145 
##           400           401           402           403           404 
## -0.1922387100 -0.0878008475 -0.0624489138 -0.2762842108  0.0024621932 
##           405           406           407           408           409 
## -0.0521510737 -0.7394339480  0.0295910486 -0.2974149703 -0.4524329524 
##           410           411           412           413           414 
## -0.0942575808 -0.2679268279 -0.1104944518 -0.5897524445  0.3438203257 
##           415           416           417           418           419 
## -0.1288860696 -0.2150139160 -0.0303903430  0.0037787687 -0.4397751792 
##           420           421           422           423           424 
## -0.6832706400 -0.4453951363 -0.1344499452 -0.8179229615 -0.4643839583 
##           425           426           427           428           429 
##  0.5094620775 -0.3647641012  0.3028159342  0.2062176388  0.1865326352 
##           430           431           432           433           434 
## -0.5719646627  0.1892242928  0.0735899417 -0.4111611070  0.2540204491 
##           435           436           437           438           439 
##  0.0494242221 -0.5904800236 -0.1388350201 -0.1684062474 -0.1357092483 
##           440           441           442           443           444 
##  0.1638572878  0.4641277285  0.9046284552 -0.5630378051 -0.2394969869 
##           445           446           447           448           449 
## -0.1976548938 -0.2692456496  0.0486816515  0.2693518653  0.1750217782 
##           450           451           452           453           454 
## -0.3212230719 -0.1564946816 -0.2735179742 -0.1829370659 -0.0562354176 
##           455           456           457           458           459 
##  0.2751055137  0.1558797044  0.1341932450 -0.3953978283  0.2066401363 
##           460           461           462           463           464 
##  0.6027321415 -0.6998549402  0.0009059039 -0.0013204327  0.0471160495 
##           465           466           467           468           469 
##  0.3756997387  0.7391826271 -0.3535968665 -0.1872812377 -0.4631693409 
##           470           471           472           473           474 
## -0.4225914438  0.2121285303  0.0406720251 -0.3903567773 -0.2813712050 
##           475           476           477           478           479 
## -0.5980437445  1.1517746897 -0.7341497203  0.2773720907  0.3208178831 
##           480           481           482           483           484 
##  0.3306062548  0.3199183622  0.5542635356 -0.8634054777 -0.1105232295 
##           485           486           487           488           489 
##  0.2386241573 -0.0339323640  0.1725353984 -0.1361991729  0.3281013347 
##           490 
## -0.0849704518
png(filename="ardl_92_forecast_acf.png", units="cm", res = 700,
    width = 20,height = 15)
grid.arrange(mod_ardl92_acf,
             mod_ardl92_meck_acf,
             mod_ardl92_hanover_acf)
dev.off()
## quartz_off_screen 
##                 2
#ARDL forecasting plots

full_cases_wastewater_weather_data_test <-
  cbind(full_cases_wastewater_weather_data_test,f_ardl92$forecasts[,2],
        f_ardl92$forecasts[,1],f_ardl92$forecasts[,3],
        f_ardl914_weather$forecasts[,2],f_ardl914_weather$forecasts[,1],
        f_ardl914_weather$forecasts[,3])

full_cases_wastewater_weather_data_meck_test <-
  cbind(full_cases_wastewater_weather_data_meck_test,f_ardl92_meck$forecasts[,2],
        f_ardl92_meck$forecasts[,1],f_ardl92_meck$forecasts[,3],
        f_ardl914_weather_meck$forecasts[,2],f_ardl914_weather_meck$forecasts[,1],
        f_ardl914_weather_meck$forecasts[,3])

full_cases_wastewater_weather_data_hanover_test <-
  cbind(full_cases_wastewater_weather_data_hanover_test,f_ardl92_hanover$forecasts[,2],
        f_ardl92_hanover$forecasts[,1],f_ardl92_hanover$forecasts[,3],
        f_ardl914_weather_hanover$forecasts[,2],f_ardl914_weather_hanover$forecasts[,1],
        f_ardl914_weather_hanover$forecasts[,3])

wake_ardl_noweather_plot <-
  full_cases_wastewater_weather_data_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_test, aes(ymin = f_ardl92$forecasts[,1], ymax = f_ardl92$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,f_ardl92$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

meck_ardl_noweather_plot <- full_cases_wastewater_weather_data_meck_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_meck_test, aes(ymin = f_ardl92_meck$forecasts[,1], ymax = f_ardl92_meck$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,f_ardl92_meck$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

hanover_ardl_noweather_plot <- full_cases_wastewater_weather_data_hanover_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_hanover_test, aes(ymin = f_ardl92_hanover$forecasts[,1], ymax = f_ardl92_hanover$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,f_ardl92_hanover$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "bottom") + ylab("")

wake_ardl_weather_plot <-
  full_cases_wastewater_weather_data_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_test, aes(ymin = f_ardl914_weather$forecasts[,1], ymax = f_ardl914_weather$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,f_ardl914_weather$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

meck_ardl_weather_plot <- full_cases_wastewater_weather_data_meck_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_meck_test, aes(ymin = f_ardl914_weather_meck$forecasts[,1], ymax = f_ardl914_weather_meck$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,f_ardl914_weather_meck$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

hanover_ardl_weather_plot <- full_cases_wastewater_weather_data_hanover_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_hanover_test, aes(ymin = f_ardl914_weather_hanover$forecasts[,1], ymax = f_ardl914_weather_hanover$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,f_ardl914_weather_hanover$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "bottom") + ylab("")

png(filename = "ardl_plots.png", res = 700, units = "cm",
    width = 20, height = 15)
grid.arrange(wake_ardl_noweather_plot,
             wake_ardl_weather_plot,
             meck_ardl_noweather_plot,
             meck_ardl_weather_plot,
             hanover_ardl_noweather_plot,
             hanover_ardl_weather_plot,
             ncol=2,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2

Distributed Lag Model

#wastewater only#
#wake

lowest_rmse_dl_wake <- Inf
best_mod_dl_wake <- NULL

for (q in seq(1,6)){
  mod <- dlm(log_mean_new_cases ~ log_viral_gene,
             data = full_cases_wastewater_weather_data_train,q=q)
  f <- forecast(mod, x= t(full_cases_wastewater_weather_data_test[,7]),h=14,
                interval = TRUE)
  forecast_acc <- rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
                       f$forecasts[,2]) #interchanged between RMSE and MAE
  if (forecast_acc<lowest_rmse_dl_wake){
    lowest_rmse_dl_wake<- forecast_acc
    best_mod_dl_wake <-mod 
  }
}


lowest_rmse_dl_wake 
## [1] 0.2338659
summary(best_mod_dl_wake) #DL(13) (lowest RMSE), DL(14) (lowest MAE)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.69819 -0.33444  0.00444  0.30269  1.62613 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -7.84578    0.23889 -32.843  < 2e-16 ***
## log_viral_gene.t  0.16035    0.04296   3.732 0.000213 ***
## log_viral_gene.1  0.05599    0.05482   1.021 0.307572    
## log_viral_gene.2  0.02580    0.05472   0.472 0.637475    
## log_viral_gene.3  0.06471    0.05463   1.185 0.236797    
## log_viral_gene.4  0.05959    0.05473   1.089 0.276826    
## log_viral_gene.5  0.07734    0.05482   1.411 0.158935    
## log_viral_gene.6  0.12980    0.04294   3.023 0.002641 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.575 on 476 degrees of freedom
## Multiple R-squared:  0.7365, Adjusted R-squared:  0.7326 
## F-statistic:   190 on 7 and 476 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 847.8488 885.4875
mod_dl13 <- dlm(log_mean_new_cases ~ log_viral_gene,
           data = full_cases_wastewater_weather_data_train,q=13)
f_dl13  <- forecast(mod_dl13, x= t(full_cases_wastewater_weather_data_test[,7]),h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
    f_dl13$forecasts) 
## [1] 0.2065803
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_dl13$forecasts)
## [1] 0.1775717
checkresiduals(mod_dl13)
##             1             2             3             4             5 
##  0.0095287990 -0.1190625530 -0.1531323134  0.0030576620  0.0136981667 
##             6             7             8             9            10 
## -0.0261142391  0.1216989061  0.0220785102 -0.1605813445  0.1785185379 
##            11            12            13            14            15 
##  0.0002528995 -0.0050411761 -0.2673081646  0.3675934026  0.0258890742 
##            16            17            18            19            20 
## -0.2323555239  0.2171339457  0.0204643421  0.1531779335  0.2517727997 
##            21            22            23            24            25 
##  0.2159750800 -0.0664740493 -0.3059228352  0.0611127354 -0.1030060694 
##            26            27            28            29            30 
## -0.0544013296  0.0469333906 -0.0452826992 -0.3949227979 -0.3531188592 
##            31            32            33            34            35 
## -0.3844135914 -0.1717731253 -0.1677140667 -0.3300973093 -0.0682872428 
##            36            37            38            39            40 
## -0.3310753380 -0.0652631266 -0.2047202392 -0.1949149682 -0.5059344021 
##            41            42            43            44            45 
## -0.3708936641 -0.1784788716 -0.1033662599  0.0841552031 -0.0395917719 
##            46            47            48            49            50 
## -0.0169014316  0.2738984334  0.4509111559  0.2785006235  0.4159804611 
##            51            52            53            54            55 
##  0.4353278508  0.5811283983  0.2265580547  0.2605957587  0.4874291118 
##            56            57            58            59            60 
##  0.3766836448  0.5413251636  0.7822674443  0.6113341442  0.6320182852 
##            61            62            63            64            65 
##  0.7296493097  0.3198245558  0.2729219371  0.1208551791  0.3032733150 
##            66            67            68            69            70 
##  0.1454792749  0.0131425149 -0.3144898677 -0.1486766130 -0.2703568530 
##            71            72            73            74            75 
## -0.0804399997  0.2007612103 -0.2139454569  0.0276957708  0.2806133265 
##            76            77            78            79            80 
## -0.1860228653 -0.4627968558 -0.2250485083  0.4136338815  0.6928904691 
##            81            82            83            84            85 
##  0.3338986126  0.2337289713  0.4859215444  0.4717119703  0.5376593731 
##            86            87            88            89            90 
##  0.7550474789  0.2957337293  0.4997706712  0.2387781890  0.2288796703 
##            91            92            93            94            95 
##  0.1762464834  0.1679184147  0.1950958591  0.1322861845  0.0056351663 
##            96            97            98            99           100 
##  0.0670879172  0.2018038871  0.2374408930  0.4707799263  0.6366294000 
##           101           102           103           104           105 
##  0.4636423640  0.4286462584  0.4379097308  0.3039290953  0.3858051744 
##           106           107           108           109           110 
##  0.1003972658  0.5755059767  0.1601121259  0.4278458633  0.2731736870 
##           111           112           113           114           115 
##  0.4419345923  0.6058745326  0.6577245491  0.8391310786  0.7369989000 
##           116           117           118           119           120 
##  0.9121247045  1.0834389217  0.8331868785  0.9874412281  0.8447164736 
##           121           122           123           124           125 
##  0.9935330582  0.9242205485  1.3084602577  0.9249033331  1.1805858542 
##           126           127           128           129           130 
##  0.6291488043  0.8845576624  1.0980101780  0.7036880288  0.9551474913 
##           131           132           133           134           135 
##  0.3246270113  0.2293898636  0.1702282967  0.2633715547  0.5195783778 
##           136           137           138           139           140 
## -1.0115844170 -0.3440971699 -1.4611202944 -0.5718723079 -0.0620514067 
##           141           142           143           144           145 
##  0.0502913088  0.3064782031 -0.5756364792 -0.7102525549 -0.5304807023 
##           146           147           148           149           150 
## -0.4409715142 -0.8216946599 -0.4266274817 -0.5141425949 -0.1281811336 
##           151           152           153           154           155 
## -0.2199366064  0.0627077707 -0.0356243330  0.1666909875  0.0037317645 
##           156           157           158           159           160 
## -0.2279486654 -0.4890579931 -0.2104378956 -0.9660052668 -0.5753503615 
##           161           162           163           164           165 
## -0.8435821343 -0.3407053536 -0.5285139637 -0.6205813727 -0.8762931537 
##           166           167           168           169           170 
## -0.9108998398 -0.6876315959 -0.5732927156 -0.8549777045 -2.5464660238 
##           171           172           173           174           175 
## -1.0196630431 -0.8949945460 -0.8511860569 -0.8429210867 -0.2374671489 
##           176           177           178           179           180 
## -0.3235065113 -0.1816112110 -0.1531358286 -0.2200797775 -0.5831020965 
##           181           182           183           184           185 
##  0.1307713552 -0.1006437630 -0.1083236903  0.1756074281  0.0444798527 
##           186           187           188           189           190 
##  0.3319336510  0.0239481241  0.3824724153  0.2569514516  0.2636273526 
##           191           192           193           194           195 
##  0.6315405112  0.3695888959  0.2119642917  0.1919769638  0.5138237890 
##           196           197           198           199           200 
##  0.4138427721  0.2867494814  0.2155279390  0.2274736345  0.0796518050 
##           201           202           203           204           205 
##  0.1060026017  0.3204923720  0.0404962344  0.0853213427  0.2438579779 
##           206           207           208           209           210 
##  0.1779288594  0.0891377187  0.0432960360  0.3103792736  0.2281222043 
##           211           212           213           214           215 
##  0.1236221727  0.1633486763  0.3235666022  0.1623906273  0.2740916076 
##           216           217           218           219           220 
##  0.5086362607  0.4734036100  0.5117131059  0.3616221789  0.6917956274 
##           221           222           223           224           225 
##  0.4847251927  0.6232416418  0.6375417112  0.6673949254  1.1145787086 
##           226           227           228           229           230 
##  0.3862205725  0.4981110260  0.4098025871  0.7143797946  0.5379146444 
##           231           232           233           234           235 
##  0.2979850152  0.2021777012 -0.5105105560 -1.9138958805 -0.1518195302 
##           236           237           238           239           240 
## -0.4888675546 -0.5724222352 -0.2815968060 -0.2388632759 -0.5393945041 
##           241           242           243           244           245 
## -0.3535085946 -0.5945209252 -0.5220520655 -0.4873440989 -0.5401129543 
##           246           247           248           249           250 
## -0.4065617273 -0.6664851319 -0.4767989824 -0.5352771914 -0.5779643305 
##           251           252           253           254           255 
## -0.5839324467 -0.4201575631 -0.6731663205 -0.9992006269 -0.3664053529 
##           256           257           258           259           260 
## -0.5702757214 -0.5596094756 -0.5228722667 -0.5166083808 -0.2655802573 
##           261           262           263           264           265 
## -0.7856158601 -0.6169771715 -0.7049805934 -0.5114925608 -0.5309703411 
##           266           267           268           269           270 
##  0.2399412232 -0.6051581838 -0.0059002330 -0.3519376422 -0.2098972172 
##           271           272           273           274           275 
##  0.3251098584  0.0843993710 -0.3616465191 -0.5206821393 -1.0270082658 
##           276           277           278           279           280 
## -0.1735646731 -0.2792434012 -0.1305361500 -2.6052706205 -1.8889804041 
##           281           282           283           284           285 
##  0.1418332207 -0.6009130439 -0.4010935176 -0.0173228884 -0.0505879051 
##           286           287           288           289           290 
## -0.2107403372 -0.0475383556  0.0022695219 -0.5074790956 -0.3666193633 
##           291           292           293           294           295 
## -0.4222073272 -0.2772176786 -0.6213116978 -0.3665745526 -0.3876669866 
##           296           297           298           299           300 
## -0.0774512187 -0.4288776958 -1.3795654658 -0.3263101456 -0.7524327828 
##           301           302           303           304           305 
## -0.0954234622 -0.4930710780 -0.2300179742 -0.1211002886 -0.3669266877 
##           306           307           308           309           310 
## -0.0209295706  0.3341295772  0.5192608572  0.9367138810  0.7860341469 
##           311           312           313           314           315 
##  0.4019857183  0.1918182390  0.1456186230 -0.3586969576  0.1490635550 
##           316           317           318           319           320 
##  0.4551800817  0.2145735941  0.5818535232  0.4892598622  0.9696658384 
##           321           322           323           324           325 
##  0.6913460519  0.6709268713  0.6574151102  0.5882864085  0.2908777784 
##           326           327           328           329           330 
##  0.3001517618  0.2613459076 -0.1673239970  0.1462799103  0.0363008613 
##           331           332           333           334           335 
##  0.1662487167  0.0390800932 -0.2331726209 -0.1887791856 -0.1968449440 
##           336           337           338           339           340 
## -0.0382003496  0.2897354781  0.1771268222  0.0940023131  0.3214349057 
##           341           342           343           344           345 
##  0.5002719619  0.8752746390  0.2933871629 -0.1518127057  1.0294021204 
##           346           347           348           349           350 
##  1.3974973867  1.5729908779  1.4411227746  1.4552885784  0.9695554381 
##           351           352           353           354           355 
##  1.0479506098  1.1558670944  1.3524316054  1.7058411420  1.4460780938 
##           356           357           358           359           360 
##  1.3356012519  1.1927786580  1.1324214981  0.4611187124  1.1398864008 
##           361           362           363           364           365 
##  0.8741525251  0.9965355980  0.6499429477  0.5565711610  0.2440844197 
##           366           367           368           369           370 
## -1.1228020026 -0.2715076753  0.6729899101  0.5860328439  0.4523365155 
##           371           372           373           374           375 
## -0.7845298391 -1.3073478210 -0.0740773196  0.5996424827 -0.0105626928 
##           376           377           378           379           380 
## -0.1102625504 -0.1950236588 -0.1132130882 -0.6834201037 -0.3669367429 
##           381           382           383           384           385 
##  0.4060755398  0.0549523976 -0.1450415496 -0.1118276624  0.1533060313 
##           386           387           388           389           390 
##  0.1062347387  0.2103637031 -0.1768461765 -0.0475146478  0.0853599094 
##           391           392           393           394           395 
## -0.2918135507 -0.3436818194 -0.2111585341 -0.0927968058  0.0417605779 
##           396           397           398           399           400 
## -0.2566887382 -0.2884695620 -0.1111722614 -0.3261228158 -0.4511201333 
##           401           402           403           404           405 
## -0.1144284548 -0.2773431750 -0.4230133363 -0.5846508273 -0.5429144957 
##           406           407           408           409           410 
## -0.3734626251 -0.9119074676 -0.4330448605 -0.7825021808 -0.7461967081 
##           411           412           413           414           415 
## -0.5831842032  0.0182994108 -0.6625402759 -0.3779988072 -0.0144497165 
##           416           417           418           419           420 
## -0.2531398401 -0.0966306430 -0.6382839044  0.0899330845 -0.3272473122 
##           421           422           423           424           425 
## -0.1910606065 -0.3627512827 -0.0012667909 -0.6182901022 -0.0532455764 
##           426           427           428           429           430 
## -0.1194188918 -0.4184490421 -0.3958815610 -0.1594862729 -0.1253070859 
##           431           432           433           434           435 
## -0.5417898680 -0.3883179832 -0.1820047742 -0.4609395941 -0.7754988071 
##           436           437           438           439           440 
## -0.7215218919 -0.9246462748 -0.3817860348 -0.7616585272 -0.6119288683 
##           441           442           443           444           445 
## -0.1375220001 -0.4939069036 -0.2968563304 -0.2997388041 -0.7178070830 
##           446           447           448           449           450 
## -0.2858385631 -0.4554618725 -0.2932202790 -0.0579412888 -0.2642574159 
##           451           452           453           454           455 
## -0.3665627261 -0.0332858150  0.1010816629  0.1257325527 -0.0526402066 
##           456           457           458           459           460 
##  0.2793252192  0.0895443023 -0.0916205798  0.0760989617 -0.1942204355 
##           461           462           463           464           465 
## -0.0607358875  0.0233958328  0.0412930454  0.2597501084 -0.2567437918 
##           466           467           468           469           470 
## -0.1226501142 -0.0914894524 -0.3034338374 -0.1470025648 -0.1449935697 
##           471           472           473           474           475 
## -0.4445127463 -0.2204801961 -0.2360937957 -0.1420129309  0.0683415345 
##           476           477 
## -0.1160366572  0.1825599262

mod_dl14 <- dlm(log_mean_new_cases ~ log_viral_gene,
           data = full_cases_wastewater_weather_data_train,q=14)
summary(mod_dl14)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.60667 -0.34908 -0.02616  0.31306  1.71001 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -8.264570   0.243490 -33.942  < 2e-16 ***
## log_viral_gene.t   0.167381   0.042160   3.970 8.33e-05 ***
## log_viral_gene.1   0.044522   0.054177   0.822   0.4116    
## log_viral_gene.2  -0.001578   0.054376  -0.029   0.9769    
## log_viral_gene.3   0.046533   0.054394   0.855   0.3927    
## log_viral_gene.4   0.043612   0.054553   0.799   0.4245    
## log_viral_gene.5   0.075028   0.054689   1.372   0.1708    
## log_viral_gene.6   0.049399   0.054840   0.901   0.3682    
## log_viral_gene.7  -0.025818   0.054896  -0.470   0.6384    
## log_viral_gene.8   0.002526   0.054849   0.046   0.9633    
## log_viral_gene.9   0.034643   0.054680   0.634   0.5267    
## log_viral_gene.10  0.032270   0.054533   0.592   0.5543    
## log_viral_gene.11  0.081860   0.054385   1.505   0.1330    
## log_viral_gene.12 -0.029197   0.054393  -0.537   0.5917    
## log_viral_gene.13  0.003275   0.054290   0.060   0.9519    
## log_viral_gene.14  0.076729   0.042232   1.817   0.0699 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5582 on 460 degrees of freedom
## Multiple R-squared:  0.7561, Adjusted R-squared:  0.7481 
## F-statistic: 95.07 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 813.5694 884.3815
f_dl14 <- forecast(mod_dl14, x= t(full_cases_wastewater_weather_data_test[,7]),
                  h=14, interval = TRUE)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_dl14$forecasts[,2]) 
## [1] 0.214038
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_dl14$forecasts[,2]) 
## [1] 0.1774498
checkresiduals(mod_dl14)
##            1            2            3            4            5            6 
## -0.151770801 -0.160174319 -0.006891305  0.012071577 -0.068008232  0.122592200 
##            7            8            9           10           11           12 
##  0.015847908 -0.178071430  0.214099747 -0.006353717 -0.012215233 -0.298606308 
##           13           14           15           16           17           18 
##  0.354637808  0.025646543 -0.260584528  0.180020051  0.013853802  0.158328818 
##           19           20           21           22           23           24 
##  0.278353673  0.210017083 -0.078877295 -0.348110739  0.055222675 -0.110048519 
##           25           26           27           28           29           30 
## -0.061257221 -0.044888949 -0.037927120 -0.391780768 -0.357219577 -0.325654025 
##           31           32           33           34           35           36 
## -0.173423164 -0.161965045 -0.331116892 -0.070850976 -0.327220624 -0.077795680 
##           37           38           39           40           41           42 
## -0.192371480 -0.199630447 -0.499721719 -0.339997397 -0.186092416 -0.112659390 
##           43           44           45           46           47           48 
##  0.046153616 -0.065795631 -0.027007380  0.272730134  0.385322871  0.284586325 
##           49           50           51           52           53           54 
##  0.408328451  0.418240641  0.607870058  0.227049609  0.244602361  0.437649300 
##           55           56           57           58           59           60 
##  0.361156912  0.546368097  0.755777260  0.480700608  0.637950909  0.761534885 
##           61           62           63           64           65           66 
##  0.367490147  0.299815161  0.125096013  0.327222264  0.264879971  0.021003598 
##           67           68           69           70           71           72 
## -0.314816009 -0.158956628 -0.277862458 -0.061712910  0.215764953 -0.300915457 
##           73           74           75           76           77           78 
##  0.034125718  0.320742606 -0.013196360 -0.464188949 -0.230510455  0.398974451 
##           79           80           81           82           83           84 
##  0.678452605  0.333376179  0.221424191  0.490438548  0.447209701  0.534268079 
##           85           86           87           88           89           90 
##  0.709815225  0.119640883  0.504932076  0.261929540  0.246888146  0.194527175 
##           91           92           93           94           95           96 
##  0.178860328  0.206730858  0.252978892  0.005364263  0.074426266  0.216287910 
##           97           98           99          100          101          102 
##  0.232817539  0.472290377  0.614895530  0.447324577  0.424654756  0.445229243 
##          103          104          105          106          107          108 
##  0.310757531  0.385827965  0.093229086  0.539775588  0.138891230  0.432588068 
##          109          110          111          112          113          114 
##  0.280276822  0.381751416  0.606126225  0.676981509  0.861906417  0.755161556 
##          115          116          117          118          119          120 
##  0.905947690  1.096363380  0.879085200  0.986306869  0.824226121  0.934572457 
##          121          122          123          124          125          126 
##  0.910965863  1.303534614  0.903807919  0.996189016  0.642240600  0.893461200 
##          127          128          129          130          131          132 
##  1.098934200  0.717405303  0.966141121  0.347449470  0.226289563  0.190708147 
##          133          134          135          136          137          138 
##  0.279640583  0.546293239 -0.972154654 -0.343724632 -1.439011674 -0.468140737 
##          139          140          141          142          143          144 
## -0.038232381  0.061734132  0.289038977 -0.648272094 -0.593070505 -0.507174123 
##          145          146          147          148          149          150 
## -0.433670957 -0.830026838 -0.441469781 -0.549782252 -0.114253460 -0.211018737 
##          151          152          153          154          155          156 
## -0.002542970 -0.025310874  0.152103161  0.021355744 -0.153498703 -0.488846098 
##          157          158          159          160          161          162 
## -0.292301718 -0.917386352 -0.565715895 -0.853973793 -0.323561354 -0.431294276 
##          163          164          165          166          167          168 
## -0.624386866 -0.973825825 -0.870499160 -0.656768083 -0.559515570 -0.817463012 
##          169          170          171          172          173          174 
## -2.439120965 -1.011462182 -0.885873005 -0.813675933 -0.818385289 -0.226150827 
##          175          176          177          178          179          180 
## -0.307792930 -0.183935955 -0.076929703 -0.215211957 -0.557805408  0.148763847 
##          181          182          183          184          185          186 
## -0.101680887 -0.117617739  0.143736499  0.020394433  0.337207754  0.045956078 
##          187          188          189          190          191          192 
##  0.332593271  0.256071911  0.247573241  0.629550849  0.509287247  0.192702377 
##          193          194          195          196          197          198 
##  0.071292443  0.565404648  0.436179259  0.301658228  0.232234299  0.393091647 
##          199          200          201          202          203          204 
##  0.079781736  0.118662310  0.338854035  0.045047977  0.088193667  0.237000094 
##          205          206          207          208          209          210 
##  0.238516762  0.086160984  0.041897754  0.315808665  0.229016112  0.125181519 
##          211          212          213          214          215          216 
##  0.151657938  0.337117793  0.162295906  0.296032161  0.514344361  0.468765657 
##          217          218          219          220          221          222 
##  0.496941398  0.346308667  0.712701681  0.474850754  0.594063627  0.649302975 
##          223          224          225          226          227          228 
##  0.655152141  1.093468567  0.364177085  0.531892550  0.388969558  0.627855968 
##          229          230          231          232          233          234 
##  0.558245460  0.300468018  0.212893475 -0.522911391 -1.841610748 -0.155285556 
##          235          236          237          238          239          240 
## -0.508681021 -0.579301985 -0.268928725 -0.189132236 -0.548010627 -0.388553086 
##          241          242          243          244          245          246 
## -0.570005532 -0.318321249 -0.476077039 -0.551777154 -0.429298466 -0.723291715 
##          247          248          249          250          251          252 
## -0.497671782 -0.548269407 -0.597052392 -0.719545452 -0.418849460 -0.672944848 
##          253          254          255          256          257          258 
## -1.010622705 -0.327623496 -0.575844083 -0.556350035 -0.523688877 -0.519657389 
##          259          260          261          262          263          264 
## -0.268375536 -0.803853303 -0.610959894 -0.710925756 -0.510139753 -0.539475414 
##          265          266          267          268          269          270 
##  0.234843138 -0.608050746 -0.027640151 -0.374300241 -0.212675738  0.334159885 
##          271          272          273          274          275          276 
##  0.076729409 -0.369232372 -0.523525111 -1.050303551 -0.205229336 -0.283065176 
##          277          278          279          280          281          282 
## -0.125127059 -2.606668860 -1.887651121  0.137770559 -0.614603483 -0.393221520 
##          283          284          285          286          287          288 
## -0.027077421 -0.031360472 -0.227814319 -0.047417728 -0.020386512 -0.539459137 
##          289          290          291          292          293          294 
## -0.383600678 -0.422027824 -0.323895183 -0.631557439 -0.365336871 -0.369111965 
##          295          296          297          298          299          300 
## -0.035236118 -0.547808614 -1.315971321 -0.329583472 -0.750814942 -0.091505927 
##          301          302          303          304          305          306 
## -0.482518346 -0.259376639 -0.122673196 -0.359898283  0.033848038  0.335394645 
##          307          308          309          310          311          312 
##  0.524354738  0.927982380  0.780829953  0.418791558  0.186413545  0.182083536 
##          313          314          315          316          317          318 
## -0.373833890  0.127315760  0.390806471  0.138672724  0.573963506  0.458075879 
##          319          320          321          322          323          324 
##  0.839090143  0.739579581  0.686325113  0.675773388  0.831578480  0.341718760 
##          325          326          327          328          329          330 
##  0.290288367  0.168494837 -0.144210596  0.144610634  0.048771048  0.254732871 
##          331          332          333          334          335          336 
##  0.042042209 -0.225549990 -0.236827529 -0.176274543 -0.038612283  0.312138062 
##          337          338          339          340          341          342 
##  0.238171523  0.136415396  0.276010470  0.540565247  0.882857955  0.296840260 
##          343          344          345          346          347          348 
## -0.137283545  1.023863671  1.419062128  1.570806375  1.494699904  1.448212011 
##          349          350          351          352          353          354 
##  0.970092989  1.044901228  1.096022120  1.360122567  1.710005296  1.493208650 
##          355          356          357          358          359          360 
##  1.339123639  1.201445618  1.143456231  0.512135705  1.140535149  0.884675146 
##          361          362          363          364          365          366 
##  1.011324742  0.652221238  0.549047199  0.263673674 -1.126485235 -0.316701781 
##          367          368          369          370          371          372 
##  0.717132716  0.585472735  0.438581794 -0.771989008 -1.290460799 -0.108444124 
##          373          374          375          376          377          378 
##  0.646418665 -0.028111462  0.033091842 -0.250391953 -0.174030337 -0.676824629 
##          379          380          381          382          383          384 
## -0.411439361  0.179887039  0.042274760 -0.135621401 -0.035189170  0.120473964 
##          385          386          387          388          389          390 
##  0.082071971  0.127908668 -0.062952369 -0.241917541  0.150867958 -0.306590869 
##          391          392          393          394          395          396 
## -0.351989173 -0.248284189 -0.229355365  0.066177156 -0.236726148 -0.360486102 
##          397          398          399          400          401          402 
## -0.117368445 -0.335625206 -0.432859367 -0.091015252 -0.352087825 -0.416276302 
##          403          404          405          406          407          408 
## -0.549777331 -0.545873481 -0.383512457 -0.921388416 -0.504774156 -0.748695127 
##          409          410          411          412          413          414 
## -0.782727437 -0.560935127 -0.008560035 -0.664322358 -0.408533211 -0.069387305 
##          415          416          417          418          419          420 
## -0.273464059 -0.132595114 -0.594053109  0.066738616 -0.340858730 -0.257368374 
##          421          422          423          424          425          426 
## -0.414811767 -0.060867515 -0.585942332 -0.240590621 -0.088740505 -0.410675912 
##          427          428          429          430          431          432 
## -0.374591384  0.066396452 -0.154500894 -0.535570467 -0.453401557 -0.172807709 
##          433          434          435          436          437          438 
## -0.458378403 -0.781191169 -0.629485851 -0.904539340 -0.437470823 -0.822157694 
##          439          440          441          442          443          444 
## -0.606324690 -0.105332875 -0.463414756 -0.232166849 -0.367055143 -0.600122567 
##          445          446          447          448          449          450 
## -0.257259980 -0.453475603 -0.286459956 -0.056093756 -0.301833108 -0.395416052 
##          451          452          453          454          455          456 
##  0.025042902  0.097045855  0.140881259 -0.036051822  0.280518057  0.096068539 
##          457          458          459          460          461          462 
## -0.029904984  0.081309895 -0.180521132 -0.067073493  0.033031739  0.025108241 
##          463          464          465          466          467          468 
##  0.216817157 -0.279499964 -0.066774088 -0.114070837 -0.288768122 -0.131827100 
##          469          470          471          472          473          474 
## -0.147972310 -0.398907266 -0.195345702 -0.192634128 -0.193318660  0.072075552 
##          475          476 
## -0.107907378  0.190390635

exp(f_dl14$forecasts[1,2])
## [1] 5.260291
exp(f_dl14$forecasts[1,1])
## [1] 1.989456
exp(f_dl14$forecasts[1,3])
## [1] 18.21049
exp(f_dl14$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_test[1,6])
## [1] -3.457252
exp(f_dl14$forecasts[7,2])
## [1] 6.063562
exp(f_dl14$forecasts[7,1])
## [1] 2.076791
exp(f_dl14$forecasts[7,3])
## [1] 17.46885
exp(f_dl14$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_test[7,6])
## [1] -0.93469
exp(f_dl14$forecasts[14,2])
## [1] 7.755135
exp(f_dl14$forecasts[14,1])
## [1] 2.46803
exp(f_dl14$forecasts[14,3])
## [1] 23.62644
exp(f_dl14$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_test[14,6])
## [1] 2.048415
mod_dl2 <- dlm(log_mean_new_cases ~ log_viral_gene,
               data = full_cases_wastewater_weather_data_train,q=2)
summary(mod_dl2)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.97051 -0.36680  0.00761  0.33854  2.00663 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -7.25687    0.25070 -28.946  < 2e-16 ***
## log_viral_gene.t  0.23468    0.04507   5.207 2.84e-07 ***
## log_viral_gene.1  0.06274    0.05888   1.066    0.287    
## log_viral_gene.2  0.23742    0.04503   5.272 2.04e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6274 on 484 degrees of freedom
## Multiple R-squared:  0.6849, Adjusted R-squared:  0.683 
## F-statistic: 350.7 on 3 and 484 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 935.9223 956.8739
f_dl2 <- forecast(mod_dl2, x= t(full_cases_wastewater_weather_data_test[,7]),
                  h=14)
rmse(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_dl2$forecasts) 
## [1] 0.2567277
mae(full_cases_wastewater_weather_data_test$log_mean_new_cases,
     f_dl2$forecasts) 
## [1] 0.218327
tsdisplay(residuals(mod_dl2))
##            1            2            3            4            5            6 
##  0.270051697  0.392966818  0.476158009  0.415748845  0.197135746  0.465756643 
##            7            8            9           10           11           12 
##  0.365134193  0.328436588  0.306786218  0.177672422  0.184030101  0.098566110 
##           13           14           15           16           17           18 
##  0.005867637  0.016270849  0.147498017  0.240383389  0.261905728  0.363702876 
##           19           20           21           22           23           24 
##  0.226906066 -0.077599345  0.316299046  0.135240553  0.207163908 -0.155827697 
##           25           26           27           28           29           30 
##  0.533140410  0.151397722  0.224921420  0.637884213  0.340086219  0.423503431 
##           31           32           33           34           35           36 
##  0.270500263  0.263643865 -0.025319008 -0.304885931  0.118496716 -0.159767758 
##           37           38           39           40           41           42 
## -0.004742163 -0.005640049 -0.132666288 -0.471258345 -0.555719617 -0.474181580 
##           43           44           45           46           47           48 
## -0.217437108 -0.110281259 -0.226478926  0.047789391 -0.240644912  0.202136736 
##           49           50           51           52           53           54 
##  0.073357057  0.068854350 -0.249763608 -0.215012896 -0.013302435  0.009360923 
##           55           56           57           58           59           60 
##  0.225891547  0.106107121  0.163011009  0.559176026  1.043331103  0.761606541 
##           61           62           63           64           65           66 
##  0.777353232  0.556032834  0.692369147  0.278467615  0.362457109  0.153941130 
##           67           68           69           70           71           72 
## -0.004039909  0.050538650  0.305321606  0.150670059  0.303042245  0.546206241 
##           73           74           75           76           77           78 
##  0.566199603  0.286710216  0.069891428 -0.481931840 -0.457480874 -0.363173383 
##           79           80           81           82           83           84 
## -0.455174495 -0.138861306 -0.421072140 -0.398024698 -0.154741923 -0.568782034 
##           85           86           87           88           89           90 
##  0.051396780  0.404503603  0.593229646  0.187685904  0.246293664  0.768840503 
##           91           92           93           94           95           96 
##  0.971739397  0.595583462  0.487198307  0.295263209  0.247191601  0.167818512 
##           97           98           99          100          101          102 
##  0.441929575 -0.050102151  0.159044587  0.012565488  0.113200502  0.044458342 
##          103          104          105          106          107          108 
##  0.091547047  0.042663336  0.113370311  0.037418062  0.167752143  0.329398388 
##          109          110          111          112          113          114 
##  0.377656898  0.553616015  0.898230235  0.669288126  0.589568706  0.482455191 
##          115          116          117          118          119          120 
##  0.278051727  0.349376460  0.027104377  0.397545451 -0.003786059  0.218715987 
##          121          122          123          124          125          126 
##  0.194446749  0.285878695  0.623632752  0.691485547  1.494795373  1.345427854 
##          127          128          129          130          131          132 
##  1.419388080  1.470312443  1.087359303  1.259729105  1.055021747  1.064779239 
##          133          134          135          136          137          138 
##  0.988724252  1.182006885  0.920494249  0.963690759  0.294742460  0.485437719 
##          139          140          141          142          143          144 
##  0.345489879  0.060806845  0.476635035 -0.049984952  0.160325418 -0.355520786 
##          145          146          147          148          149          150 
## -0.287642492 -0.020865928 -1.418223137 -0.550539502 -1.666149567 -0.678315446 
##          151          152          153          154          155          156 
## -0.130523792  0.200301142  0.514793178 -0.591599357 -0.750730105 -0.778953428 
##          157          158          159          160          161          162 
## -0.616725269 -0.624296040 -0.287427807 -0.449411914 -0.349569682 -0.491121589 
##          163          164          165          166          167          168 
## -0.409269675 -0.432584269  0.232313580  0.021988546 -0.273046069 -0.691193476 
##          169          170          171          172          173          174 
## -0.587789326 -1.524106476 -1.110829482 -1.287740886 -0.841628967 -1.000483182 
##          175          176          177          178          179          180 
## -1.022017330 -1.395151934 -1.302321336 -1.299590451 -1.069169379 -1.425359426 
##          181          182          183          184          185          186 
## -2.970508001 -1.324335329 -1.193829999 -1.042744001 -1.042986365 -0.357452952 
##          187          188          189          190          191          192 
## -0.374422873  0.038573643  0.139257128 -0.121674589 -0.536092697 -0.255566064 
##          193          194          195          196          197          198 
## -0.365745394  0.139626215  0.401815614  0.278745047  0.156316024 -0.165588212 
##          199          200          201          202          203          204 
## -0.313004304 -0.354086961 -0.207277703  0.202083575  0.062430764 -0.135871772 
##          205          206          207          208          209          210 
## -0.209746572 -0.003618880 -0.050244154  0.005435248 -0.085656864  0.145511342 
##          211          212          213          214          215          216 
## -0.026051798  0.070372763  0.226212897 -0.060794842 -0.023578114  0.110740594 
##          217          218          219          220          221          222 
##  0.154817354  0.057539365  0.077035923  0.261975619  0.186807488  0.226747491 
##          223          224          225          226          227          228 
##  0.241652912  0.462981501  0.222490894  0.381470345  0.568158687  0.557901698 
##          229          230          231          232          233          234 
##  0.895083680  0.680714438  0.998050609  0.661825738  0.765809943  0.622840918 
##          235          236          237          238          239          240 
##  0.651075849  1.077073831  0.332076338  0.459665677  0.437390480  0.749655186 
##          241          242          243          244          245          246 
##  0.549148519  0.210065909 -0.527997274 -1.221193801 -2.422708327 -0.450573917 
##          247          248          249          250          251          252 
## -0.610231048 -0.810124779 -0.410089833 -0.294943033 -0.104015337  0.215190791 
##          253          254          255          256          257          258 
## -0.197778700 -0.025612119 -0.220297484 -0.248057534 -0.146142645 -0.481908547 
##          259          260          261          262          263          264 
## -0.263836756 -0.468252405 -0.397555413 -0.549355059 -0.402858334 -0.662335034 
##          265          266          267          268          269          270 
## -1.023264211 -0.306699790 -0.476700972 -0.385776165 -0.307324128 -0.346582205 
##          271          272          273          274          275          276 
## -0.178964351 -0.747413067 -0.534665919 -0.569661437 -0.291709850 -0.258698643 
##          277          278          279          280          281          282 
##  0.458341603 -0.468685588  0.080495990 -0.249485727 -0.095336840  0.513537199 
##          283          284          285          286          287          288 
##  0.208293724 -0.277121510 -0.521821984 -1.042768919 -0.165823650 -0.230022553 
##          289          290          291          292          293          294 
## -0.005820125 -2.367461701 -1.709146529  0.440407142 -0.384529244 -0.240862362 
##          295          296          297          298          299          300 
##  0.206936085 -0.039865951  0.206531557  0.105202687  0.071631134 -0.509327943 
##          301          302          303          304          305          306 
## -0.422332798 -0.355607411 -0.111628639 -0.564440837 -0.290865502 -0.659051495 
##          307          308          309          310          311          312 
## -0.204004155 -0.613664115 -1.470805973 -0.337429022 -0.867471546 -0.248035290 
##          313          314          315          316          317          318 
## -0.649431820 -0.403257322 -0.184479290 -0.437246758  0.085518248  0.610152696 
##          319          320          321          322          323          324 
##  0.914942979  2.006628712  1.675226452  0.947022561  0.383144247 -0.574389567 
##          325          326          327          328          329          330 
## -0.961788695 -0.298681841  0.545110400  0.293846282  0.257424952  0.071789745 
##          331          332          333          334          335          336 
##  0.129096393  0.075839849  0.374635320  0.563532411  0.680596171  0.213420359 
##          337          338          339          340          341          342 
##  0.160522090 -0.077909402 -0.613335222 -0.001095000 -0.278531529 -0.017642249 
##          343          344          345          346          347          348 
## -0.178417829 -0.420773912 -0.291468919 -0.401117168 -0.219447323 -0.064118479 
##          349          350          351          352          353          354 
## -0.151559737 -0.012681382  0.229603534  0.665857221  0.952500469  0.322748254 
##          355          356          357          358          359          360 
## -0.226422194  0.934061962  1.389460730  1.560308073  1.313311072  1.309050418 
##          361          362          363          364          365          366 
##  0.747963580  0.819291582  0.929753311  1.188165087  1.624131450  1.385343656 
##          367          368          369          370          371          372 
##  1.409390626  1.126650114  0.974262725  0.371340617  1.173783794  0.973385100 
##          373          374          375          376          377          378 
##  1.339886225  0.651051308  0.656613577 -0.358291420 -1.563803253 -0.593937891 
##          379          380          381          382          383          384 
##  0.725376915  0.908688097  1.341421727 -0.074975313 -0.721341795  0.009820264 
##          385          386          387          388          389          390 
##  0.970983858  0.285689868  0.895267358  0.161574217  0.666534641 -0.474948308 
##          391          392          393          394          395          396 
## -0.088619239  0.678475095  0.338018657  0.550802916  0.576091181  0.529517191 
##          397          398          399          400          401          402 
##  0.741653385  0.476480850  0.373961863  0.323679712  0.369879128  0.162712000 
##          403          404          405          406          407          408 
## -0.058518666 -0.104060802 -0.112017719  0.050951802 -0.131415338  0.104563859 
##          409          410          411          412          413          414 
##  0.053276379 -0.092777352 -0.452444396 -0.071862668 -0.144286972 -0.265169778 
##          415          416          417          418          419          420 
## -0.167530888 -0.119852311 -0.010592448 -0.596637775 -0.291780834 -0.572431409 
##          421          422          423          424          425          426 
## -0.524476802 -0.381992912  0.601516456 -0.253288587  0.828225503  0.941562372 
##          427          428          429          430          431          432 
##  0.332658448  0.210824121 -1.239570787 -0.167003226 -0.464695430  0.023787661 
##          433          434          435          436          437          438 
## -0.111491675 -0.294240202 -0.896646299 -0.745566537 -0.651616954 -0.503223045 
##          439          440          441          442          443          444 
## -0.347567598  0.081167828 -0.165192619 -0.553835019 -0.717276363 -0.204684507 
##          445          446          447          448          449          450 
## -0.802116513 -1.243301663 -1.141765058 -1.282322973 -0.579209912 -0.923876275 
##          451          452          453          454          455          456 
## -0.718991611 -0.369982407 -0.777400663 -0.405229759 -0.424967900 -0.763923450 
##          457          458          459          460          461          462 
## -0.322593050 -0.708692810 -0.594475349 -0.393429235 -0.511584162 -0.576924044 
##          463          464          465          466          467          468 
## -0.138878504  0.056506591  0.211944673 -0.181155200  0.361967467  0.121323581 
##          469          470          471          472          473          474 
## -0.021802897  0.120158880 -0.415688929 -0.265938907 -0.301744682 -0.019734757 
##          475          476          477          478          479          480 
##  0.262074152 -0.402834214 -0.220238248 -0.491042184 -0.527245371 -0.371250466 
##          481          482          483          484          485          486 
## -0.296393898 -0.496860186 -0.352586221 -0.231493563 -0.207379903  0.184858043 
##          487          488 
##  0.034526012  0.308302624

#mecklenburg

lowest_rmse_dl_meck <- Inf
best_mod_dl_meck <- NULL

for (q in seq(1,14)){
  mod <- dlm(log_mean_new_cases ~ log_viral_gene,
             data = full_cases_wastewater_weather_data_meck_train,q=q)
  f <- forecast(mod, x= t(full_cases_wastewater_weather_data_meck_test[,8]),h=14,
                interval = TRUE)
  forecast_acc <- rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
                       f$forecasts[,2])
  if (forecast_acc<lowest_rmse_dl_meck){
    lowest_rmse_dl_meck<- forecast_acc
    best_mod_dl_meck <-mod 
  }
}


lowest_rmse_dl_meck #0.212, 0.149
## [1] 0.2122282
summary(best_mod_dl_meck) #DL(2)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9644 -0.3551  0.0139  0.3342  2.4752 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -11.18759    0.36463 -30.682  < 2e-16 ***
## log_viral_gene.t   0.28237    0.06590   4.285 2.20e-05 ***
## log_viral_gene.1   0.11153    0.08830   1.263    0.207    
## log_viral_gene.2   0.35381    0.06552   5.400 1.04e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6201 on 484 degrees of freedom
## Multiple R-squared:  0.6879, Adjusted R-squared:  0.686 
## F-statistic: 355.7 on 3 and 484 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##       AIC      BIC
## 1 924.456 945.4076
checkresiduals(best_mod_dl_meck)
##             1             2             3             4             5 
##  0.0878210139  0.3800073610  0.3744189744  0.2224762173  0.1289952774 
##             6             7             8             9            10 
##  0.3618215647  0.4311459317  0.5173828389  0.2498647545  0.2459964098 
##            11            12            13            14            15 
##  0.3671665497  0.3266017253  0.3230277124  0.2624211421  0.2614478592 
##            16            17            18            19            20 
##  0.1417386391  0.3738787833  0.6582855570  0.5884310684  0.3534665893 
##            21            22            23            24            25 
##  0.1703357025  0.4721527786  0.4170098038  0.2797150439  0.3413641694 
##            26            27            28            29            30 
##  0.6073583852  0.8027948260  0.7231762347  0.6441895867  0.3269096718 
##            31            32            33            34            35 
##  0.0206588211 -0.1856444081  0.2832685352  0.0914426569 -0.1949623021 
##            36            37            38            39            40 
## -0.2878551098 -0.4609823114 -0.1439375704  0.0299054615 -0.1398242289 
##            41            42            43            44            45 
## -0.2390741351  0.0815095854 -0.0614045209 -0.1451269686  0.0422653507 
##            46            47            48            49            50 
## -0.1948615136  0.2187748466 -0.3628210458 -0.8106227426 -0.3819471623 
##            51            52            53            54            55 
## -0.3600536513 -0.0717598059 -0.1082864631 -0.2197058396  0.4153064938 
##            56            57            58            59            60 
## -0.0754918747  0.2162053565 -0.8626604131 -0.8104160698 -0.2226433699 
##            61            62            63            64            65 
## -0.8937618943  0.7697239950  1.1378736596  1.8131536285  1.8099988631 
##            66            67            68            69            70 
##  2.1154907044  2.2811008252  2.4752055463  1.1691885586  0.9079300138 
##            71            72            73            74            75 
## -0.3459719917  0.1508996029 -0.1559081837  0.3243754992  0.4766392039 
##            76            77            78            79            80 
##  0.5125433514  0.2646871902  0.1023414631 -0.0045949843 -0.2685873156 
##            81            82            83            84            85 
## -0.0848307333 -0.0385798826  0.1713571528 -0.2861106668  0.1031875596 
##            86            87            88            89            90 
##  0.0005460578  0.2571983271  0.0022882660  0.0337302033  0.4748384700 
##            91            92            93            94            95 
##  0.0980440317  0.2486419174  0.0314852899 -0.0838581626 -0.0083080292 
##            96            97            98            99           100 
## -0.1064370833  0.0751896500 -0.0045140084  0.3669699467  0.1785183070 
##           101           102           103           104           105 
## -0.2392410495  0.3561993871 -0.2358444839  0.6051792825  0.3242977958 
##           106           107           108           109           110 
##  0.2066500784  0.4170491229  0.1480623223  0.3974642028  0.7600343121 
##           111           112           113           114           115 
##  0.5112332846  0.4762998428  0.1014601380 -0.3412409831 -0.0754742015 
##           116           117           118           119           120 
## -0.4360419950 -0.4342811288 -0.2410305631 -0.3245172836 -0.3545028704 
##           121           122           123           124           125 
## -0.4460961412 -0.8274934435 -0.5853430418 -0.2160234932  0.0042513993 
##           126           127           128           129           130 
## -0.5306383062  0.4990229976 -0.2123754706  0.5598229002  0.5147167433 
##           131           132           133           134           135 
##  1.2401141341  0.5114043986  0.2890544648 -0.4902832576 -0.6081426472 
##           136           137           138           139           140 
## -0.2788478780 -0.7055956017 -0.6896501004  0.4039368977 -0.0619616724 
##           141           142           143           144           145 
## -0.4967710096  0.1963164685  0.4889487757  0.3820255160  0.7651740161 
##           146           147           148           149           150 
##  0.4743381872  0.2302727008  1.3486237340  0.1114170252 -0.3847607964 
##           151           152           153           154           155 
## -1.0716595037  0.2806894893 -0.6050143017 -0.7885182086 -1.0754080648 
##           156           157           158           159           160 
## -0.5830989873 -0.4349648860 -0.9086270042 -0.3056534439  0.1298147136 
##           161           162           163           164           165 
##  0.1624682117  0.3245212557 -0.1536432871 -0.5535588781 -0.6189787223 
##           166           167           168           169           170 
## -0.4108601145 -0.5681596992 -1.4326406932 -0.8976950851 -0.8651557972 
##           171           172           173           174           175 
## -0.8362828045 -1.3785007585 -0.6222170298 -0.2004651997 -1.0149715355 
##           176           177           178           179           180 
## -0.8708239731 -1.5060745936 -0.3779234002 -0.2299975404 -0.3552922621 
##           181           182           183           184           185 
## -0.6076610236 -0.3802574530 -0.5520942136 -0.2714904264  0.2083385043 
##           186           187           188           189           190 
##  0.2179882865  0.4190906633  0.0971735397 -0.3055417006  0.0115603021 
##           191           192           193           194           195 
## -0.1790631591 -0.0785164719 -0.1424524813 -0.2519345208  0.1824475177 
##           196           197           198           199           200 
## -0.0954619473 -0.0275230892  0.2401983648  0.2950517375  0.2542087654 
##           201           202           203           204           205 
##  0.2906126846  0.0216548838  0.1572355912 -0.0032882137  0.4562520516 
##           206           207           208           209           210 
##  0.4997724798  0.1509238282  0.3064077888  0.0954090561 -0.1257807871 
##           211           212           213           214           215 
## -0.3790084180 -0.4873693084 -0.2132264846 -0.1684593812 -0.3042875778 
##           216           217           218           219           220 
##  0.0440491844  0.1687394112  0.1832216449  0.5964707056  0.5211846336 
##           221           222           223           224           225 
##  0.3715915728  0.0069657893 -0.1513280427  0.0195084419  0.1272107407 
##           226           227           228           229           230 
##  0.2952290498  0.3600791085  0.5297388492  0.4871792087  0.6639204248 
##           231           232           233           234           235 
##  0.5330634345  0.2279048217 -0.0368473728  0.0740038324  0.3061452091 
##           236           237           238           239           240 
##  0.1833652147  0.1207739869  0.2244091347  0.0254349373 -0.2508751356 
##           241           242           243           244           245 
##  0.0136821816 -0.1360793038 -0.2601655023 -0.3476054021 -1.2323267744 
##           246           247           248           249           250 
## -0.3398661453 -0.2451239748 -0.2517688738 -0.1206268626 -0.0958757677 
##           251           252           253           254           255 
## -0.1541163964  0.0316403291 -0.0752231665 -0.3848131114 -0.1969297984 
##           256           257           258           259           260 
## -0.1349276013 -0.3193680544 -0.5517259829 -0.2429448878 -0.3643660672 
##           261           262           263           264           265 
## -0.3677536969 -0.1497141476 -0.1501457512 -0.1263816550 -0.2766910847 
##           266           267           268           269           270 
## -0.0176196611 -0.2765009728  0.2425566290  0.0573853432  0.0141216007 
##           271           272           273           274           275 
## -0.1920277090 -0.6006628496 -0.2905867876 -0.3377693707 -0.6555412071 
##           276           277           278           279           280 
## -0.5262880744 -0.0433097513 -0.3281500723  0.0802360317  0.1366874503 
##           281           282           283           284           285 
## -0.0289287982  0.2007907266  0.0484089886  0.1708490841  0.1663320321 
##           286           287           288           289           290 
## -0.1860313771 -0.2109119629 -0.4932488442 -0.3531522267 -1.4064850922 
##           291           292           293           294           295 
## -1.1532720126 -0.1330494429 -0.0033190306 -0.3634634168 -0.3550048457 
##           296           297           298           299           300 
## -0.0590388066 -0.5021739726  0.0842670234  0.3518348266  0.1736946037 
##           301           302           303           304           305 
## -0.1638730742 -0.2010613353 -1.1360668040 -0.5676220975 -0.3985995027 
##           306           307           308           309           310 
## -0.1872866473  0.0270202768  0.0691583115 -0.2129174022 -0.0062470079 
##           311           312           313           314           315 
##  0.0228906731  0.0234590896  0.0584204958  0.1027359395 -0.0140892831 
##           316           317           318           319           320 
##  0.2097155041  0.3508939425  0.3665036540  0.2863343792  0.4489783540 
##           321           322           323           324           325 
##  0.3085233704  0.2358179270  0.5033533275  0.4879103557 -0.0438926432 
##           326           327           328           329           330 
##  0.5388422080  1.2792482022  0.8340736175  0.6710077933  0.3941027350 
##           331           332           333           334           335 
##  0.3010289929  0.3252155671  0.5232905003  0.1661305372 -0.2726705901 
##           336           337           338           339           340 
## -0.2012880855 -0.3898541799 -0.5022516428 -0.4913300972 -0.1074613641 
##           341           342           343           344           345 
## -0.0348003293  0.1048526419  0.4156613099  0.2720342299  0.4898100485 
##           346           347           348           349           350 
##  0.6294764796  0.5540817584  0.5137900114  0.3317648112  0.5339611122 
##           351           352           353           354           355 
##  0.7459580002  1.0497901207  1.1976668526  0.9924092017  0.2652819621 
##           356           357           358           359           360 
##  1.3961882323  1.7268734658  1.8154003469  1.8537873092  1.9696396100 
##           361           362           363           364           365 
##  1.6321456305  1.3734573904  1.8372771601  1.3748796356  1.2851071147 
##           366           367           368           369           370 
##  0.6766798514  0.2939523010  0.3821186827 -0.2109714895  0.0716801269 
##           371           372           373           374           375 
##  0.6389329311  0.5144747557  0.7280385878  0.7234862218  0.7490553586 
##           376           377           378           379           380 
##  0.4305749749 -1.2557039193 -0.1819290503  0.5152550651  0.5164116182 
##           381           382           383           384           385 
##  0.5267622758  0.2408123778 -0.1993893489  0.2066629680  0.3977152039 
##           386           387           388           389           390 
##  0.2047798242  0.1569723710  0.0813437916  0.0459501676 -0.1416181932 
##           391           392           393           394           395 
##  0.0191113542  0.5500275194  0.1987352021  0.7297226462  0.4890137605 
##           396           397           398           399           400 
##  0.6558494116  0.3726109228  0.3207934548  1.0012190410  0.8052336735 
##           401           402           403           404           405 
##  1.2363070464  1.1523518150  0.8791922614  0.6545646600 -0.1976625334 
##           406           407           408           409           410 
##  0.4774693193  0.1086378007  0.2138499238  0.2113394376  0.2320715188 
##           411           412           413           414           415 
##  0.2024586817  0.2476425476  0.5958574009  0.6097536271  0.8810846180 
##           416           417           418           419           420 
##  0.7249281713  0.8731739179 -0.0377808311 -1.3841622185 -0.3319225939 
##           421           422           423           424           425 
## -0.7941100884 -0.7520911894 -0.4967659227 -0.7468647428 -1.1265872635 
##           426           427           428           429           430 
##  0.1382658891 -0.8296419141 -1.0117849156 -1.2488893370 -1.8075341017 
##           431           432           433           434           435 
## -1.0969729383 -0.8732810942 -1.1328422403 -1.0851673760 -0.8410658029 
##           436           437           438           439           440 
##  0.5757437670 -0.1673485431 -0.5650527492 -0.8413939749 -1.0939810172 
##           441           442           443           444           445 
## -0.7240712727 -1.1301022485 -0.8080670738 -0.6315737562 -0.8368956126 
##           446           447           448           449           450 
## -1.0987831437 -0.2008766537 -1.2226962410  0.1027709802 -0.2926703197 
##           451           452           453           454           455 
## -0.5646493951 -0.8817429341 -0.6307754280 -0.8522008641 -0.4673554302 
##           456           457           458           459           460 
## -0.8584311360 -0.3759321478 -0.2457658142 -0.5640702694 -1.0387994037 
##           461           462           463           464           465 
## -0.3312933428  0.0818168001 -0.0499004269  0.1569463603 -0.1985874568 
##           466           467           468           469           470 
## -0.9463877453 -1.3420041139 -1.9643629170 -1.0568113313 -0.9979023375 
##           471           472           473           474           475 
## -0.9311086544 -0.7326924603 -0.8322180793 -1.0314967906 -0.5993223386 
##           476           477           478           479           480 
## -0.1502704111 -0.4834551844 -0.3747059760 -0.4652754436 -0.6305169163 
##           481           482           483           484           485 
## -0.4924727479 -0.6181290957 -0.5199200960 -0.6593226135 -0.5830868700 
##           486           487           488 
## -0.5728043461 -0.4415182273 -0.2856220900

mod_dl13_meck <- dlm(log_mean_new_cases ~ log_viral_gene,
               data = full_cases_wastewater_weather_data_meck_train,q=13)
summary(mod_dl13_meck)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.73309 -0.33153 -0.01849  0.29939  1.86748 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -12.69622    0.36233 -35.041  < 2e-16 ***
## log_viral_gene.t    0.20943    0.06059   3.456 0.000598 ***
## log_viral_gene.1    0.10162    0.08119   1.252 0.211377    
## log_viral_gene.2    0.01814    0.08120   0.223 0.823341    
## log_viral_gene.3    0.13229    0.08118   1.630 0.103882    
## log_viral_gene.4   -0.01279    0.08116  -0.158 0.874832    
## log_viral_gene.5    0.06239    0.08117   0.769 0.442470    
## log_viral_gene.6    0.02190    0.08071   0.271 0.786203    
## log_viral_gene.7    0.07913    0.08052   0.983 0.326239    
## log_viral_gene.8    0.04023    0.08058   0.499 0.617882    
## log_viral_gene.9    0.02790    0.08057   0.346 0.729321    
## log_viral_gene.10   0.04824    0.08062   0.598 0.549946    
## log_viral_gene.11   0.06230    0.08064   0.772 0.440224    
## log_viral_gene.12  -0.02368    0.08020  -0.295 0.767941    
## log_viral_gene.13   0.07544    0.05998   1.258 0.209093    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5586 on 462 degrees of freedom
## Multiple R-squared:  0.7522, Adjusted R-squared:  0.7447 
## F-statistic: 100.2 on 14 and 462 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 814.8224 881.5027
f_dl13_meck <- forecast(mod_dl13_meck, 
                       x= t(full_cases_wastewater_weather_data_meck_test[,8]),
                       h=14)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_dl13_meck$forecasts) 
## [1] 0.3192566
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_dl13_meck$forecasts)
## [1] 0.2851718
checkresiduals(mod_dl13_meck)
##             1             2             3             4             5 
##  0.1154434640  0.2270340292  0.1352352965  0.1303615803  0.0510685951 
##             6             7             8             9            10 
##  0.1315203487  0.4052983027  0.3961927409  0.1406918438 -0.0184856821 
##            11            12            13            14            15 
##  0.4249214051  0.3649271924  0.0748429973  0.1864538032  0.4611759268 
##            16            17            18            19            20 
##  0.5471971296  0.5095830728  0.2561405400  0.0984354697  0.1226892835 
##            21            22            23            24            25 
## -0.1545718404  0.3455899352  0.1406183373 -0.0769016649 -0.1017796615 
##            26            27            28            29            30 
## -0.3005851425 -0.0631893579  0.0373683413 -0.1490635157 -0.2746711349 
##            31            32            33            34            35 
## -0.0641437710 -0.1942494164 -0.4105111857 -0.1294883922 -0.3816067611 
##            36            37            38            39            40 
##  0.0513366351 -0.5058665684 -0.8671666250 -0.2086301484 -0.2621130355 
##            41            42            43            44            45 
## -0.2622951365 -0.2198464113 -0.3815319399  0.2145794599 -0.2888187268 
##            46            47            48            49            50 
## -0.0886916200 -1.0618606282 -0.9294641841 -0.3059824930 -0.9673356253 
##            51            52            53            54            55 
##  0.4786226459  0.8888609172  0.7051449141  0.9121232008  1.1884105847 
##            56            57            58            59            60 
##  1.4883461331  1.7798923890  0.8876330788  0.7898089310  0.6592105489 
##            61            62            63            64            65 
##  0.9386759679  0.8681499302  1.1211544072  1.3971857848  1.1992074642 
##            66            67            68            69            70 
##  0.8661658767  0.6851140186  0.4517411439  0.0743119166  0.2713356005 
##            71            72            73            74            75 
##  0.0868381477  0.2690146590 -0.1817420173  0.3041275630  0.1631036238 
##            76            77            78            79            80 
##  0.2806294629  0.0252346283  0.0168765856  0.3523197629 -0.0422787474 
##            81            82            83            84            85 
## -0.1059178104 -0.1600370295  0.0222768006  0.0374899407 -0.0372072494 
##            86            87            88            89            90 
##  0.0734926953  0.0494457678  0.2035631336  0.1302101722 -0.2156379315 
##            91            92            93            94            95 
##  0.3077742906 -0.2629356738  0.5856755520  0.2645592814  0.0681282556 
##            96            97            98            99           100 
##  0.3590388903  0.0235103853  0.2842293473  0.6780099823  0.4135143607 
##           101           102           103           104           105 
##  0.3955151594  0.0418059724 -0.3018305016  0.3035450425 -0.1784841753 
##           106           107           108           109           110 
## -0.1281243703 -0.0584270735 -0.0959087697 -0.3638632057 -0.2499124165 
##           111           112           113           114           115 
## -0.6312873074 -0.4452861115 -0.1259963552 -0.0294754723 -0.6486568944 
##           116           117           118           119           120 
## -0.3122164823 -0.7359902264 -0.2215079793 -0.1153478731  0.6795349476 
##           121           122           123           124           125 
##  0.1773442653  0.0600701709 -0.1889507606 -0.3097936116  0.2051331663 
##           126           127           128           129           130 
## -0.3375305716 -0.1608711229  0.6703004590  0.2102639144 -0.6676801351 
##           131           132           133           134           135 
##  0.2081887957  0.7348189866  0.5480498377  0.4667766296  0.3528875268 
##           136           137           138           139           140 
##  0.1350074863  1.3052752368  0.3190712057 -0.1178076547 -0.0528254602 
##           141           142           143           144           145 
##  0.5757944381  0.0161492465 -0.3346466580 -0.3315275269  0.0103178648 
##           146           147           148           149           150 
##  0.1385654475 -0.4081662802  0.0213600537  0.2785856680  0.4381865192 
##           151           152           153           154           155 
##  0.2902639854 -0.0483963354 -0.4370624796 -0.4269600594  0.0587650638 
##           156           157           158           159           160 
## -0.2269002839 -1.0032598113 -1.0994138191 -0.3511850892 -0.6148724060 
##           161           162           163           164           165 
## -1.1621377955 -0.4810709251 -0.2644705917 -1.0132644077 -0.8380323525 
##           166           167           168           169           170 
## -1.1137161570 -0.0942174231 -0.0597049633 -0.0533479763 -0.6249979097 
##           171           172           173           174           175 
## -0.3451893475 -0.4822907257 -0.1887738421  0.2871146244  0.3261707724 
##           176           177           178           179           180 
##  0.5703240099  0.4102482499 -0.0426223866  0.3533020389  0.1348035726 
##           181           182           183           184           185 
##  0.2319533437  0.1583455810  0.0174336244  0.6699546788  0.2408587594 
##           186           187           188           189           190 
##  0.3036190525  0.4429280681  0.4440327257  0.3936354323  0.4215217713 
##           191           192           193           194           195 
##  0.4317954116  0.4012271112  0.2458079710  0.4927100052  0.5252357846 
##           196           197           198           199           200 
##  0.1693024748  0.3159285588  0.3943514267  0.0976188288 -0.1418879854 
##           201           202           203           204           205 
## -0.1206977194  0.0140737600 -0.0033403409 -0.2246424804  0.0078717060 
##           206           207           208           209           210 
##  0.0574706898  0.0284970374  0.2295973935  0.1771223515  0.0475649165 
##           211           212           213           214           215 
## -0.2409293082 -0.2003294810 -0.0364154866  0.1410262664  0.1262449786 
##           216           217           218           219           220 
##  0.2703524923  0.4000392222  0.3211932411  0.4699224823  0.4082027723 
##           221           222           223           224           225 
##  0.1130797367  0.0616690343  0.0619215483  0.3199954451  0.1522828022 
##           226           227           228           229           230 
##  0.0673150840  0.1732020126 -0.0015922747 -0.2434029080 -0.0248030554 
##           231           232           233           234           235 
## -0.1748995223 -0.2931907728 -0.2937878893 -1.2233721708 -0.2888865323 
##           236           237           238           239           240 
## -0.2057416870 -0.2455276237 -0.1868723618 -0.1894794816 -0.4054396305 
##           241           242           243           244           245 
## -0.1864621667 -0.3464268712 -0.5808044949 -0.4494625225 -0.3465372735 
##           246           247           248           249           250 
## -0.5219881631 -0.6795818991 -0.4072144894 -0.4740392080 -0.6594986917 
##           251           252           253           254           255 
## -0.3662173995 -0.4302400871 -0.3796893668 -0.6536580513 -0.3203183959 
##           256           257           258           259           260 
## -0.5431666474 -0.0297483332 -0.1993672491 -0.1643963831 -0.3633078744 
##           261           262           263           264           265 
## -0.5932981077 -0.3165781789 -0.2997600422 -0.7190597063 -0.5570845699 
##           266           267           268           269           270 
## -0.1699240588 -0.4766515524 -0.3550780963 -0.1978323973 -0.3591741417 
##           271           272           273           274           275 
## -0.1369848419 -0.2807230695 -0.0647239886 -0.0303891845 -0.2330683495 
##           276           277           278           279           280 
## -0.2497993460 -0.4343118668 -0.3944618074 -1.3742536351 -1.1321409061 
##           281           282           283           284           285 
## -0.1243059746  0.0649356156 -0.3209308316 -0.2910931237 -0.0452687746 
##           286           287           288           289           290 
## -0.5063281870  0.0258508009  0.2788148558 -0.0926920569 -0.2558142424 
##           291           292           293           294           295 
## -0.2592924281 -0.7674038540 -0.3604768079 -0.1815794261 -0.0356889821 
##           296           297           298           299           300 
## -0.0714876491  0.0005602339 -0.2676355729 -0.1260995957 -0.0846714591 
##           301           302           303           304           305 
## -0.0767350582  0.0235402762  0.1404842677 -0.0114305765  0.2999126993 
##           306           307           308           309           310 
##  0.2258628904  0.3472683135  0.2338500739  0.4008778843  0.2800907637 
##           311           312           313           314           315 
##  0.2093030035  0.5031227312  0.4051220630 -0.1119382661  0.4839754811 
##           316           317           318           319           320 
##  1.2218486676  0.7935483343  0.7361709530  0.5115400839  0.7365557525 
##           321           322           323           324           325 
##  0.6348473258  0.9072978629  0.4875919613  0.2602226728  0.2201507143 
##           326           327           328           329           330 
##  0.0455444754  0.0192482907 -0.1023649558  0.1234451759  0.1496027279 
##           331           332           333           334           335 
## -0.0396850055  0.2993906555  0.1228609058  0.3750343526  0.4439116287 
##           336           337           338           339           340 
##  0.4474401233  0.4478643209  0.5314217771  0.6426221564  0.9396866941 
##           341           342           343           344           345 
##  1.0807834314  1.2795425619  0.9919503664  0.2390952487  1.3257238647 
##           346           347           348           349           350 
##  1.6582117010  1.7382339251  1.8288656699  1.8674786547  1.5405359199 
##           351           352           353           354           355 
##  1.2659866030  1.7227066897  1.4210764575  1.3857186361  1.4492874470 
##           356           357           358           359           360 
##  0.8665485288  0.9204658828  0.5620756731  0.3497518935  0.8571256600 
##           361           362           363           364           365 
##  0.6038809885  0.5919902011  0.4861884069  0.3972610781  0.1601925395 
##           366           367           368           369           370 
## -1.6965490615 -0.5914658017  0.2450075816  0.1778329267  0.2552411733 
##           371           372           373           374           375 
## -0.0273428988 -0.4965941114 -0.1713912626  0.0617665349 -0.1057698974 
##           376           377           378           379           380 
## -0.1642314360 -0.2433329984 -0.2806006880 -0.4566661748 -0.3713376271 
##           381           382           383           384           385 
##  0.1451190608 -0.1602316576  0.0279101382 -0.0583899848  0.1007041774 
##           386           387           388           389           390 
## -0.1191529073 -0.1870452845  0.5200688855  0.3996637267  0.4361169431 
##           391           392           393           394           395 
##  0.5711442316  0.4109962261  0.2472779619 -0.2182240622  0.4120938091 
##           396           397           398           399           400 
##  0.1408284270  0.0033971556  0.1218508480  0.1446594079  0.0857092846 
##           401           402           403           404           405 
##  0.2392272010  0.4805895070  0.4877681182  0.2615586832  0.2613297400 
##           406           407           408           409           410 
##  0.5213670447 -0.3031257319 -1.0884887150 -0.1297666285 -0.4855471198 
##           411           412           413           414           415 
## -0.6305870869 -0.3004546768 -0.6195914988 -1.0812247936  0.0832105854 
##           416           417           418           419           420 
## -0.8033589899 -1.0265155575 -1.0091989937 -1.7330922522 -0.9714270095 
##           421           422           423           424           425 
## -0.7782122871 -1.0861968333 -1.1029338163 -0.8271216764  0.1727848249 
##           426           427           428           429           430 
## -0.4099969807 -0.7376174045 -0.9101061610 -0.6802461435 -0.4035309941 
##           431           432           433           434           435 
## -0.6957691148 -0.4932027928 -0.2714978903 -0.5559209501 -0.9065429830 
##           436           437           438           439           440 
## -0.0865147144 -1.0772312836  0.1963166592  0.0128402944 -0.4330519803 
##           441           442           443           444           445 
## -0.7126733203 -0.4859006563 -0.6370821092 -0.3007303924 -0.3710199612 
##           446           447           448           449           450 
## -0.2396837812 -0.0267083452 -0.4286597886 -0.8771774033 -0.1981861173 
##           451           452           453           454           455 
##  0.2051874546  0.1103586582  0.2454179238 -0.0198330161 -0.6883519250 
##           456           457           458           459           460 
## -0.6038275469 -1.2853307839 -0.4356281543 -0.4561568625 -0.6124627996 
##           461           462           463           464           465 
## -0.5174878498 -0.7535609522 -0.9973528513 -0.7892364594 -0.3424722037 
##           466           467           468           469           470 
## -0.6082043324 -0.5059209299 -0.6381575460 -0.7128357867 -0.5859581855 
##           471           472           473           474           475 
## -0.5339275520 -0.4603546985 -0.5398331452 -0.4226189531 -0.4406174862 
##           476           477 
## -0.3922388130 -0.2259694150

mod_dl14_meck <- dlm(log_mean_new_cases ~ log_viral_gene,
                     data = full_cases_wastewater_weather_data_meck_train,q=14)
summary(mod_dl14_meck)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.52782 -0.33597 -0.01237  0.30179  1.98147 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -12.790812   0.363314 -35.206  < 2e-16 ***
## log_viral_gene.t    0.210989   0.060327   3.497 0.000515 ***
## log_viral_gene.1    0.110288   0.080908   1.363 0.173509    
## log_viral_gene.2    0.012731   0.080872   0.157 0.874982    
## log_viral_gene.3    0.121994   0.080927   1.507 0.132380    
## log_viral_gene.4   -0.006169   0.080849  -0.076 0.939209    
## log_viral_gene.5    0.055441   0.080898   0.685 0.493492    
## log_viral_gene.6    0.013397   0.080883   0.166 0.868518    
## log_viral_gene.7    0.085842   0.080384   1.068 0.286125    
## log_viral_gene.8    0.045452   0.080255   0.566 0.571438    
## log_viral_gene.9    0.018783   0.080295   0.234 0.815143    
## log_viral_gene.10   0.052275   0.080281   0.651 0.515278    
## log_viral_gene.11   0.060375   0.080365   0.751 0.452877    
## log_viral_gene.12  -0.024241   0.080316  -0.302 0.762928    
## log_viral_gene.13  -0.056401   0.079950  -0.705 0.480886    
## log_viral_gene.14   0.147705   0.059744   2.472 0.013786 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5561 on 460 degrees of freedom
## Multiple R-squared:  0.755,  Adjusted R-squared:  0.747 
## F-statistic: 94.51 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 809.8517 880.6638
f_dl14_meck <- forecast(mod_dl14_meck, 
                        x= t(full_cases_wastewater_weather_data_meck_test[,8]),
                        h=14,interval = TRUE)
rmse(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
     f_dl14_meck$forecasts[,2]) 
## [1] 0.3135576
mae(full_cases_wastewater_weather_data_meck_test$log_mean_new_cases,
    f_dl14_meck$forecasts[,2]) 
## [1] 0.2853209
checkresiduals(mod_dl14_meck)
##            1            2            3            4            5            6 
##  0.083673414  0.203715871  0.108310564  0.045605367  0.110362522  0.393824828 
##            7            8            9           10           11           12 
##  0.283817235  0.228569589 -0.026232304  0.407629201  0.361447059  0.052547076 
##           13           14           15           16           17           18 
##  0.178001206  0.472462403  0.552468387  0.433889572  0.248862452  0.077375063 
##           19           20           21           22           23           24 
##  0.100255268 -0.155851852  0.383149327  0.144611745 -0.141688388 -0.114885685 
##           25           26           27           28           29           30 
## -0.305414029 -0.073615076  0.035925731 -0.252943021 -0.244670279  0.087310528 
##           31           32           33           34           35           36 
## -0.197463150 -0.414951938 -0.149099555 -0.381173178  0.105180877 -0.507462204 
##           37           38           39           40           41           42 
## -0.927093891 -0.222960778 -0.261467804 -0.278147642 -0.227185504 -0.378278193 
##           43           44           45           46           47           48 
##  0.159035080 -0.273853376 -0.092976634 -1.069741022 -0.948109530 -0.306405589 
##           49           50           51           52           53           54 
## -0.863064403  0.480081475  0.778805931  0.711123070  0.895631949  1.175119854 
##           55           56           57           58           59           60 
##  1.470857538  1.673620842  0.877213084  0.798740972  0.606666470  0.926610972 
##           61           62           63           64           65           66 
##  0.822451538  1.109450802  1.085787869  1.182070764  0.857658253  0.731467011 
##           67           68           69           70           71           72 
##  0.488390679  0.108432878  0.324193401  0.536438611  0.288742072 -0.172080777 
##           73           74           75           76           77           78 
##  0.300727820  0.168724681  0.274726918  0.031480759  0.049598504  0.373607464 
##           79           80           81           82           83           84 
## -0.002866966 -0.108282393 -0.176879302  0.005139536  0.039720451  0.006671769 
##           85           86           87           88           89           90 
##  0.082309573  0.012106944  0.195568556  0.117092948 -0.226234898  0.304477535 
##           91           92           93           94           95           96 
## -0.389882718  0.613438625  0.412712740  0.057138967  0.352902740  0.008560608 
##           97           98           99          100          101          102 
##  0.278130924  0.602149876  0.430990641  0.419864616  0.032364319 -0.316782862 
##          103          104          105          106          107          108 
##  0.283565605 -0.175242847 -0.158905309 -0.037251591 -0.110680855 -0.362177583 
##          109          110          111          112          113          114 
## -0.254869641 -0.628388144 -0.436689816 -0.150006003  0.007474222 -0.479450536 
##          115          116          117          118          119          120 
## -0.311955557 -0.741149972 -0.227838354 -0.128741053  0.581673961  0.229427592 
##          121          122          123          124          125          126 
##  0.065854073 -0.227517218 -0.322209225  0.165986256 -0.351316573 -0.412853967 
##          127          128          129          130          131          132 
##  0.703709198  0.141136814 -0.637472279  0.209881186  0.749356210  0.590173817 
##          133          134          135          136          137          138 
##  0.642901903  0.368915261  0.176590063  1.292981938  0.296486825 -0.152654063 
##          139          140          141          142          143          144 
## -0.047852413  0.396782323  0.042512824 -0.140314907 -0.315514338 -0.152180505 
##          145          146          147          148          149          150 
##  0.139575873 -0.356172563  0.071344091  0.330071292  0.465744800  0.647180537 
##          151          152          153          154          155          156 
## -0.240704446 -0.433002635 -0.418524658  0.178443669 -0.202221510 -0.988534265 
##          157          158          159          160          161          162 
## -1.104346842 -0.378046737 -0.599136381 -1.160978629 -0.504870864 -0.218204950 
##          163          164          165          166          167          168 
## -0.997848618 -0.847337339 -0.970523182 -0.058223360 -0.066176988 -0.276425474 
##          169          170          171          172          173          174 
## -0.394131347 -0.348757251 -0.486810964 -0.184599029  0.181121759  0.341363720 
##          175          176          177          178          179          180 
##  0.597635822  0.554918616 -0.046330493  0.344939888  0.135983889  0.157789623 
##          181          182          183          184          185          186 
##  0.172392220  0.035975707  0.697702454  0.251142866  0.314330712  0.462649289 
##          187          188          189          190          191          192 
##  0.554286428  0.404283953  0.432232194  0.454116679  0.405956061  0.254142897 
##          193          194          195          196          197          198 
##  0.506486369  0.639235343  0.173128478  0.317428907  0.357183148  0.097078915 
##          199          200          201          202          203          204 
## -0.145247699 -0.115962461  0.152087391 -0.005357401 -0.211596058 -0.050151630 
##          205          206          207          208          209          210 
##  0.056863765  0.037783605  0.240534616  0.288189768  0.043790436 -0.240101494 
##          211          212          213          214          215          216 
## -0.157707365 -0.056896908  0.125865046  0.109479264  0.211576331  0.394874826 
##          217          218          219          220          221          222 
##  0.324116687  0.368418890  0.408429453  0.104188411  0.054861850  0.144561498 
##          223          224          225          226          227          228 
##  0.321135690  0.158932473  0.006441350  0.161730015 -0.014472257 -0.247536562 
##          229          230          231          232          233          234 
## -0.058038082 -0.174460627 -0.280441817 -0.218221623 -1.236400020 -0.303947449 
##          235          236          237          238          239          240 
## -0.212385134 -0.267329530 -0.187324705 -0.174693433 -0.379732549 -0.198600813 
##          241          242          243          244          245          246 
## -0.355265131 -0.588452008 -0.407572550 -0.358375338 -0.520762780 -0.680479784 
##          247          248          249          250          251          252 
## -0.425806851 -0.490049514 -0.671583984 -0.429451793 -0.435406359 -0.394451973 
##          253          254          255          256          257          258 
## -0.648419830 -0.339791705 -0.554080852 -0.044116526 -0.192096899 -0.179573145 
##          259          260          261          262          263          264 
## -0.367558237 -0.675288139 -0.331304639 -0.313166651 -0.728111349 -0.611805831 
##          265          266          267          268          269          270 
## -0.168956313 -0.458199417 -0.367018295 -0.211525045 -0.361337641 -0.143816302 
##          271          272          273          274          275          276 
## -0.244640382 -0.077397245 -0.029399448 -0.277802980 -0.267612959 -0.450998155 
##          277          278          279          280          281          282 
## -0.408262589 -1.499252712 -1.129477922 -0.112450865  0.039643067 -0.324413390 
##          283          284          285          286          287          288 
## -0.290525830 -0.041874533 -0.459955189  0.027960488  0.297860449 -0.112198979 
##          289          290          291          292          293          294 
## -0.267310416 -0.273467438 -0.776781816 -0.342723799 -0.185990398 -0.008966314 
##          295          296          297          298          299          300 
## -0.075112821 -0.005481946 -0.272122270 -0.122574727 -0.174129075 -0.065889338 
##          301          302          303          304          305          306 
##  0.047714726  0.298739868 -0.025017908  0.289811610  0.217793082  0.239615121 
##          307          308          309          310          311          312 
##  0.239571458  0.414151066  0.257634125  0.204833149  0.506767865  0.406909890 
##          313          314          315          316          317          318 
## -0.057258669  0.482213312  1.231821470  0.721636017  0.728648391  0.500368238 
##          319          320          321          322          323          324 
##  0.731164422  0.642727538  0.907181007  0.502277235  0.234604449  0.227290927 
##          325          326          327          328          329          330 
##  0.041763187  0.034718302 -0.090580190  0.144835121  0.191011942  0.117674808 
##          331          332          333          334          335          336 
##  0.296869566  0.125979818  0.381965504  0.525056634  0.443657128  0.454052683 
##          337          338          339          340          341          342 
##  0.586118672  0.628964184  0.923735461  1.072776733  1.185790856  0.998244704 
##          343          344          345          346          347          348 
##  0.253340933  1.331491217  1.652599841  1.737726257  1.830308586  1.981473344 
##          349          350          351          352          353          354 
##  1.532152911  1.272594809  1.682241025  1.404727965  1.354108842  1.432703322 
##          355          356          357          358          359          360 
##  0.857938160  0.905015563  0.588338824  0.397328873  0.853828081  0.610746309 
##          361          362          363          364          365          366 
##  0.621299482  0.476321744  0.409339184  0.197701568 -1.422533507 -0.616534030 
##          367          368          369          370          371          372 
##  0.223464548  0.300899504  0.100463617 -0.048853561 -0.501220205 -0.265002792 
##          373          374          375          376          377          378 
##  0.038305076 -0.124335803 -0.181012347 -0.264522309 -0.295319384 -0.454387699 
##          379          380          381          382          383          384 
## -0.375987394  0.119548942 -0.173341685  0.010694421 -0.135396856  0.088129302 
##          385          386          387          388          389          390 
## -0.122878115 -0.224254237  0.491590967  0.390222335  0.413354790  0.507118521 
##          391          392          393          394          395          396 
##  0.390756165  0.224829384 -0.395754231  0.394496872  0.131855036 -0.010801313 
##          397          398          399          400          401          402 
##  0.093833413  0.137200493  0.081621831  0.060338738  0.480498679  0.507988234 
##          403          404          405          406          407          408 
##  0.272410677  0.368974209  0.515291904 -0.319724897 -1.206895697 -0.137937622 
##          409          410          411          412          413          414 
## -0.489106405 -0.636590369 -0.282526529 -0.618754014 -1.073271939 -0.120288440 
##          415          416          417          418          419          420 
## -0.796112808 -1.013106913 -0.995653753 -1.527820192 -0.975681290 -0.760436347 
##          421          422          423          424          425          426 
## -1.135932461 -1.105016648 -0.818919288  0.181468417 -0.490653826 -0.725625459 
##          427          428          429          430          431          432 
## -0.903965703 -0.619550178 -0.408564854 -0.701227412 -0.496222007 -0.290672830 
##          433          434          435          436          437          438 
## -0.547162448 -0.890290220 -0.228505158 -1.064244357  0.213677325  0.033515382 
##          439          440          441          442          443          444 
## -0.217047684 -0.711160301 -0.465666053 -0.649149701 -0.309713911 -0.370760781 
##          445          446          447          448          449          450 
## -0.226032639 -0.077406740 -0.409509028 -0.842848432 -0.124986960  0.201279365 
##          451          452          453          454          455          456 
##  0.119362479  0.250029094 -0.013946757 -0.695203647 -0.479920168 -1.362543319 
##          457          458          459          460          461          462 
## -0.437744273 -0.441604822 -0.580462180 -0.474034803 -0.742117255 -0.958541231 
##          463          464          465          466          467          468 
## -0.784336697 -0.334692619 -0.592994945 -0.270884088 -0.603632500 -0.728979911 
##          469          470          471          472          473          474 
## -0.588428646 -0.612960962 -0.473513648 -0.557718658 -0.430461009 -0.501864842 
##          475          476 
## -0.389870632 -0.205309993

exp(f_dl14_meck$forecasts[1,2])
## [1] 4.929395
exp(f_dl14_meck$forecasts[1,1])
## [1] 1.590384
exp(f_dl14_meck$forecasts[1,3])
## [1] 14.41146
exp(f_dl14_meck$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_meck_test[1,7])
## [1] 1.903267
exp(f_dl14_meck$forecasts[7,2])
## [1] 5.127421
exp(f_dl14_meck$forecasts[7,1])
## [1] 1.602196
exp(f_dl14_meck$forecasts[7,3])
## [1] 13.41248
exp(f_dl14_meck$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_meck_test[7,7])
## [1] 1.272963
exp(f_dl14_meck$forecasts[14,2])
## [1] 4.977879
exp(f_dl14_meck$forecasts[14,1])
## [1] 1.613458
exp(f_dl14_meck$forecasts[14,3])
## [1] 13.98632
exp(f_dl14_meck$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_meck_test[14,7])
## [1] 1.957939
#New Hanover

lowest_rmse_dl_hanover <- Inf
best_mod_dl_hanover <- NULL

for (q in seq(1,14)){
  mod <- dlm(log_mean_new_cases ~ log_viral_gene,
             data = full_cases_wastewater_weather_data_hanover_train,q=q)
  f <- forecast(mod, x= t(full_cases_wastewater_weather_data_hanover_test[,7]),h=14)
  forecast_acc <- mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
                       f$forecasts)
  if (forecast_acc<lowest_rmse_dl_hanover){
    lowest_rmse_dl_hanover<- forecast_acc
    best_mod_dl_hanover <-mod 
  }
}


lowest_rmse_dl_hanover #0.581,0.449
## [1] 0.4490573
summary(best_mod_dl_hanover) #DL(14)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.92770 -0.54283  0.01895  0.56104  1.54669 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -8.275542   0.323175 -25.607   <2e-16 ***
## log_viral_gene.t   0.162004   0.059146   2.739   0.0064 ** 
## log_viral_gene.1   0.035519   0.080932   0.439   0.6610    
## log_viral_gene.2  -0.008743   0.081057  -0.108   0.9141    
## log_viral_gene.3   0.056927   0.082394   0.691   0.4900    
## log_viral_gene.4   0.109439   0.082429   1.328   0.1849    
## log_viral_gene.5   0.008675   0.083221   0.104   0.9170    
## log_viral_gene.6   0.019041   0.083370   0.228   0.8194    
## log_viral_gene.7   0.022755   0.083408   0.273   0.7851    
## log_viral_gene.8   0.091259   0.083419   1.094   0.2745    
## log_viral_gene.9  -0.050251   0.083424  -0.602   0.5472    
## log_viral_gene.10  0.020449   0.082742   0.247   0.8049    
## log_viral_gene.11  0.091420   0.082722   1.105   0.2697    
## log_viral_gene.12  0.013366   0.081523   0.164   0.8698    
## log_viral_gene.13 -0.012672   0.081525  -0.155   0.8765    
## log_viral_gene.14  0.031479   0.059567   0.528   0.5974    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7457 on 460 degrees of freedom
## Multiple R-squared:  0.6205, Adjusted R-squared:  0.6082 
## F-statistic: 50.15 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 1089.257 1160.069
tsdisplay(residuals(best_mod_dl_hanover))
##            1            2            3            4            5            6 
##  0.747671788  0.307641508  0.378269471  0.473802219  0.619452137  0.711640715 
##            7            8            9           10           11           12 
##  0.784764229  0.851181386  0.587009244  0.494110837  0.452150818  0.244520123 
##           13           14           15           16           17           18 
##  1.122428761  0.583604437  0.912896054  0.215617235  0.691798693  0.600442412 
##           19           20           21           22           23           24 
##  0.524673916  0.873202989  1.163039524  1.087161000  0.647565642  0.586902041 
##           25           26           27           28           29           30 
##  0.787283320  0.752887310  0.961409396  0.688481219  0.380161119  0.573329284 
##           31           32           33           34           35           36 
##  0.443081736  0.358261805  0.027539317  0.293750417 -0.150229915  0.271321532 
##           37           38           39           40           41           42 
##  0.122643563 -0.191472098  0.266413213 -0.044373382  0.366898659  0.576119311 
##           43           44           45           46           47           48 
##  0.673770853  0.442499814  0.353702994  0.518073738  0.721501191  1.016752586 
##           49           50           51           52           53           54 
##  0.968241689  0.797826891  0.564507013  1.403061846  1.371784098  1.189120163 
##           55           56           57           58           59           60 
##  0.831064425  1.273177601  0.966582768  0.844717559  1.023075623  0.807142418 
##           61           62           63           64           65           66 
##  1.018402414  1.150247042  1.277706262  1.143776243  0.986930140  1.396903771 
##           67           68           69           70           71           72 
##  1.453890134  1.287967088  1.252050855  1.127398886  1.351576932  1.109953607 
##           73           74           75           76           77           78 
##  1.439402548  0.826177275  0.798481506  0.559887259  0.878814915  0.460054856 
##           79           80           81           82           83           84 
##  0.618189988  0.354178901  0.800934551  0.644800810  0.272745138  0.556436014 
##           85           86           87           88           89           90 
##  0.698699805  0.765819983  0.676171855  0.678981448  0.541217929  0.878639863 
##           91           92           93           94           95           96 
##  0.480107399 -0.567074083  0.835251043  0.443114773  0.798032497  0.475458293 
##           97           98           99          100          101          102 
##  0.929089500  1.151674303 -0.365146410  1.092744210  0.958868595  0.975938609 
##          103          104          105          106          107          108 
##  0.727230555  0.924896703  0.191074765  0.630066217  0.830326618  0.300237101 
##          109          110          111          112          113          114 
##  0.431653877 -0.470709044  0.587993289  0.713496419  0.170954979  0.864112086 
##          115          116          117          118          119          120 
##  0.419857150  0.941106815  0.304678257  0.619942958  0.000665306  0.049049003 
##          121          122          123          124          125          126 
##  0.532278958 -0.483855497 -0.438958916  0.642657182  0.719306023  0.083623883 
##          127          128          129          130          131          132 
##  0.142308364 -0.547859467  0.679362968 -0.466719588  0.386998573 -0.101381340 
##          133          134          135          136          137          138 
##  0.172972328  0.335315478 -0.049409208 -0.011413372  0.034182159 -0.186190837 
##          139          140          141          142          143          144 
## -0.077516488  0.206178558  0.216607175 -0.276560341 -0.139998257 -0.212428313 
##          145          146          147          148          149          150 
##  0.010084704  0.058939187  0.568749793  0.651043913 -0.375174713 -0.182188461 
##          151          152          153          154          155          156 
## -0.276029586 -0.002888015  0.087513971  0.351314577  0.403895340 -0.027424995 
##          157          158          159          160          161          162 
## -0.004155429  0.089282066  0.116703756 -0.267717173  0.613447150  0.103826154 
##          163          164          165          166          167          168 
## -0.584795668 -0.797745125 -0.923444321 -0.080705315 -0.362428882 -0.193055134 
##          169          170          171          172          173          174 
## -0.022549817 -0.528540196  0.017414671 -0.263990755 -0.635740946 -1.006383371 
##          175          176          177          178          179          180 
## -0.719630746  0.041794680 -0.919307011 -1.240639400  0.153899524 -0.275186941 
##          181          182          183          184          185          186 
##  0.255706766 -0.622651534 -1.010647658 -1.643449840 -0.706496720 -0.554271322 
##          187          188          189          190          191          192 
##  0.085125471 -0.153401590  0.196242232  0.466242979  0.101916302  0.132891233 
##          193          194          195          196          197          198 
## -0.058730517  0.226849953  0.092034675  0.046388800  0.320837843  0.197634154 
##          199          200          201          202          203          204 
##  0.041543478 -0.084591895 -0.031534623  0.376878462  0.083499436  0.131331710 
##          205          206          207          208          209          210 
##  0.125611940  0.160134357 -0.025976111 -0.147004743  0.102707357 -0.132043750 
##          211          212          213          214          215          216 
##  0.142449049  0.222082014 -0.095722886 -0.086447134  0.193431092  0.047369108 
##          217          218          219          220          221          222 
## -0.236510673 -0.011232245  0.630867557  0.965698775  0.842951821  0.980329919 
##          223          224          225          226          227          228 
##  0.405603758 -0.081259496  0.020476796  0.198995617  0.539530739  0.695241902 
##          229          230          231          232          233          234 
##  0.505353089  0.103543071  0.084199783 -0.060773234 -0.687262335  0.223897168 
##          235          236          237          238          239          240 
##  0.158265282 -0.097488959 -0.063130119 -0.320686525 -0.111602996  0.223043923 
##          241          242          243          244          245          246 
##  0.069286926 -0.075827417 -0.195341651  0.007382603 -0.155997697 -0.340909232 
##          247          248          249          250          251          252 
## -0.313957509 -0.278744127 -0.331574026 -0.279316587 -0.669261650 -0.511348060 
##          253          254          255          256          257          258 
## -0.543592370 -0.567951602 -0.649225670 -0.795434355 -1.068118107 -0.536898969 
##          259          260          261          262          263          264 
## -0.781653867 -1.378215169 -0.642796987 -0.695479334 -1.133816241 -0.705216939 
##          265          266          267          268          269          270 
## -0.833183745 -0.481082101 -0.949364510 -0.666934935 -0.729533596 -0.670244008 
##          271          272          273          274          275          276 
## -0.459775767 -0.312795933 -0.683795518 -0.958265199 -0.710554591 -0.461965690 
##          277          278          279          280          281          282 
## -1.927703353 -1.860354405 -1.788623255 -0.580177525 -1.364166882 -1.264233742 
##          283          284          285          286          287          288 
## -1.405559426 -0.994684914 -1.846049333 -1.169577240 -1.522802689 -1.370957154 
##          289          290          291          292          293          294 
## -1.488322029 -0.898873445 -1.182528558 -1.869183395 -1.303331789 -0.838020471 
##          295          296          297          298          299          300 
## -0.783640584 -0.184541374 -0.804762372 -0.845862677 -0.639393492 -0.073034083 
##          301          302          303          304          305          306 
##  0.505106704 -0.414760325 -0.152886163  0.011629339  0.175510337  0.008692918 
##          307          308          309          310          311          312 
## -0.466941362 -0.303730275  0.212979426 -0.354538102 -0.630846915 -0.610448748 
##          313          314          315          316          317          318 
## -1.751952004 -1.272773337 -0.374306882  0.290963727  0.046446696 -0.214981661 
##          319          320          321          322          323          324 
## -0.545341288 -0.332527264 -0.469964243 -0.455682589 -0.018678859 -0.219464918 
##          325          326          327          328          329          330 
## -0.255516321  0.299562431  0.462774125 -0.712331746  0.204310949  0.378700005 
##          331          332          333          334          335          336 
##  0.236595198  0.131521713  0.023084524  0.309143039  0.393019139  0.514684828 
##          337          338          339          340          341          342 
##  0.709765643  0.815688120  0.422194522  0.927938263  1.026089235  0.119582660 
##          343          344          345          346          347          348 
## -0.400948873  0.618876707  1.283429018  1.428341833  1.416771338  1.546686323 
##          349          350          351          352          353          354 
##  0.844014152  0.520292465  0.964667125  1.468735348  1.440001829  1.088767430 
##          355          356          357          358          359          360 
##  1.381710323  1.285408988  0.927875505  0.875859442  1.146208669  1.053124920 
##          361          362          363          364          365          366 
##  0.992434881  1.103931611  1.235399280  0.792817833  0.536064069  1.022193226 
##          367          368          369          370          371          372 
##  1.369988900  1.386744406  1.145528674  0.111164949 -0.122587151  0.403142012 
##          373          374          375          376          377          378 
##  0.917741437  0.574358546  0.414126633  0.390922055  0.409998132 -0.047921889 
##          379          380          381          382          383          384 
##  0.051743407  0.346350681  0.219834318  0.183649959  0.003918620 -0.087689089 
##          385          386          387          388          389          390 
## -0.443879508 -0.368255592 -0.233817769 -0.134835441 -0.394255333 -0.319883083 
##          391          392          393          394          395          396 
## -0.218698343 -0.910699733 -0.465972929 -0.585333047 -0.694596814 -0.586108231 
##          397          398          399          400          401          402 
## -0.738585332 -0.679538577 -1.118610524 -0.420766359 -0.638185474 -0.714555809 
##          403          404          405          406          407          408 
## -0.585440652 -0.563505308 -0.878102761 -1.280603695 -1.191681077 -0.934102963 
##          409          410          411          412          413          414 
## -1.596363594 -1.557809531 -0.722601433 -1.255975639 -0.770097847 -0.727531531 
##          415          416          417          418          419          420 
## -0.518165467 -1.293445438 -0.778215674 -0.565879295 -0.856861437 -0.400860881 
##          421          422          423          424          425          426 
## -0.440988575 -0.959018978 -0.689239873 -0.689466316 -0.532029981 -0.278728684 
##          427          428          429          430          431          432 
##  0.266328315  0.783612944 -0.362063857 -0.402825267 -0.403869896 -0.366283882 
##          433          434          435          436          437          438 
## -0.063202851  0.165020700  0.151977809 -0.472627095 -0.542580764 -0.820343633 
##          439          440          441          442          443          444 
## -0.583075855 -0.521115134 -0.403670372 -0.526089683 -0.566016826 -1.104702309 
##          445          446          447          448          449          450 
## -0.806441249 -0.082094347 -0.979519688 -0.878787309 -0.900340594 -0.696046618 
##          451          452          453          454          455          456 
## -0.459637216  0.026626468 -0.546385035 -0.882709570 -1.394742797 -1.432171235 
##          457          458          459          460          461          462 
## -0.853846173 -0.715528872 -1.211671395 -1.384127172 -1.836427085 -0.407158641 
##          463          464          465          466          467          468 
## -1.607260055 -0.880169779 -0.844179745 -0.666642593 -0.660131954 -0.278294858 
##          469          470          471          472          473          474 
## -1.266666170 -1.041434986 -0.633569678 -0.426953761 -0.315162411 -0.559516785 
##          475          476 
## -0.196315618 -0.496435946

mod_dl14_hanover <- dlm(log_mean_new_cases ~ log_viral_gene,
           data = full_cases_wastewater_weather_data_hanover_train,q=14)
summary(mod_dl14_hanover)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.92770 -0.54283  0.01895  0.56104  1.54669 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -8.275542   0.323175 -25.607   <2e-16 ***
## log_viral_gene.t   0.162004   0.059146   2.739   0.0064 ** 
## log_viral_gene.1   0.035519   0.080932   0.439   0.6610    
## log_viral_gene.2  -0.008743   0.081057  -0.108   0.9141    
## log_viral_gene.3   0.056927   0.082394   0.691   0.4900    
## log_viral_gene.4   0.109439   0.082429   1.328   0.1849    
## log_viral_gene.5   0.008675   0.083221   0.104   0.9170    
## log_viral_gene.6   0.019041   0.083370   0.228   0.8194    
## log_viral_gene.7   0.022755   0.083408   0.273   0.7851    
## log_viral_gene.8   0.091259   0.083419   1.094   0.2745    
## log_viral_gene.9  -0.050251   0.083424  -0.602   0.5472    
## log_viral_gene.10  0.020449   0.082742   0.247   0.8049    
## log_viral_gene.11  0.091420   0.082722   1.105   0.2697    
## log_viral_gene.12  0.013366   0.081523   0.164   0.8698    
## log_viral_gene.13 -0.012672   0.081525  -0.155   0.8765    
## log_viral_gene.14  0.031479   0.059567   0.528   0.5974    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7457 on 460 degrees of freedom
## Multiple R-squared:  0.6205, Adjusted R-squared:  0.6082 
## F-statistic: 50.15 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 1089.257 1160.069
f_dl14_hanover <- forecast(mod_dl14_hanover,
                          x= t(full_cases_wastewater_weather_data_hanover_test[,7]),
                          h=14,interval = TRUE)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_dl14_hanover$forecasts[,2])
## [1] 0.5808326
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_dl14_hanover$forecasts[,2])
## [1] 0.4490573
checkresiduals(mod_dl14_hanover)
##            1            2            3            4            5            6 
##  0.747671788  0.307641508  0.378269471  0.473802219  0.619452137  0.711640715 
##            7            8            9           10           11           12 
##  0.784764229  0.851181386  0.587009244  0.494110837  0.452150818  0.244520123 
##           13           14           15           16           17           18 
##  1.122428761  0.583604437  0.912896054  0.215617235  0.691798693  0.600442412 
##           19           20           21           22           23           24 
##  0.524673916  0.873202989  1.163039524  1.087161000  0.647565642  0.586902041 
##           25           26           27           28           29           30 
##  0.787283320  0.752887310  0.961409396  0.688481219  0.380161119  0.573329284 
##           31           32           33           34           35           36 
##  0.443081736  0.358261805  0.027539317  0.293750417 -0.150229915  0.271321532 
##           37           38           39           40           41           42 
##  0.122643563 -0.191472098  0.266413213 -0.044373382  0.366898659  0.576119311 
##           43           44           45           46           47           48 
##  0.673770853  0.442499814  0.353702994  0.518073738  0.721501191  1.016752586 
##           49           50           51           52           53           54 
##  0.968241689  0.797826891  0.564507013  1.403061846  1.371784098  1.189120163 
##           55           56           57           58           59           60 
##  0.831064425  1.273177601  0.966582768  0.844717559  1.023075623  0.807142418 
##           61           62           63           64           65           66 
##  1.018402414  1.150247042  1.277706262  1.143776243  0.986930140  1.396903771 
##           67           68           69           70           71           72 
##  1.453890134  1.287967088  1.252050855  1.127398886  1.351576932  1.109953607 
##           73           74           75           76           77           78 
##  1.439402548  0.826177275  0.798481506  0.559887259  0.878814915  0.460054856 
##           79           80           81           82           83           84 
##  0.618189988  0.354178901  0.800934551  0.644800810  0.272745138  0.556436014 
##           85           86           87           88           89           90 
##  0.698699805  0.765819983  0.676171855  0.678981448  0.541217929  0.878639863 
##           91           92           93           94           95           96 
##  0.480107399 -0.567074083  0.835251043  0.443114773  0.798032497  0.475458293 
##           97           98           99          100          101          102 
##  0.929089500  1.151674303 -0.365146410  1.092744210  0.958868595  0.975938609 
##          103          104          105          106          107          108 
##  0.727230555  0.924896703  0.191074765  0.630066217  0.830326618  0.300237101 
##          109          110          111          112          113          114 
##  0.431653877 -0.470709044  0.587993289  0.713496419  0.170954979  0.864112086 
##          115          116          117          118          119          120 
##  0.419857150  0.941106815  0.304678257  0.619942958  0.000665306  0.049049003 
##          121          122          123          124          125          126 
##  0.532278958 -0.483855497 -0.438958916  0.642657182  0.719306023  0.083623883 
##          127          128          129          130          131          132 
##  0.142308364 -0.547859467  0.679362968 -0.466719588  0.386998573 -0.101381340 
##          133          134          135          136          137          138 
##  0.172972328  0.335315478 -0.049409208 -0.011413372  0.034182159 -0.186190837 
##          139          140          141          142          143          144 
## -0.077516488  0.206178558  0.216607175 -0.276560341 -0.139998257 -0.212428313 
##          145          146          147          148          149          150 
##  0.010084704  0.058939187  0.568749793  0.651043913 -0.375174713 -0.182188461 
##          151          152          153          154          155          156 
## -0.276029586 -0.002888015  0.087513971  0.351314577  0.403895340 -0.027424995 
##          157          158          159          160          161          162 
## -0.004155429  0.089282066  0.116703756 -0.267717173  0.613447150  0.103826154 
##          163          164          165          166          167          168 
## -0.584795668 -0.797745125 -0.923444321 -0.080705315 -0.362428882 -0.193055134 
##          169          170          171          172          173          174 
## -0.022549817 -0.528540196  0.017414671 -0.263990755 -0.635740946 -1.006383371 
##          175          176          177          178          179          180 
## -0.719630746  0.041794680 -0.919307011 -1.240639400  0.153899524 -0.275186941 
##          181          182          183          184          185          186 
##  0.255706766 -0.622651534 -1.010647658 -1.643449840 -0.706496720 -0.554271322 
##          187          188          189          190          191          192 
##  0.085125471 -0.153401590  0.196242232  0.466242979  0.101916302  0.132891233 
##          193          194          195          196          197          198 
## -0.058730517  0.226849953  0.092034675  0.046388800  0.320837843  0.197634154 
##          199          200          201          202          203          204 
##  0.041543478 -0.084591895 -0.031534623  0.376878462  0.083499436  0.131331710 
##          205          206          207          208          209          210 
##  0.125611940  0.160134357 -0.025976111 -0.147004743  0.102707357 -0.132043750 
##          211          212          213          214          215          216 
##  0.142449049  0.222082014 -0.095722886 -0.086447134  0.193431092  0.047369108 
##          217          218          219          220          221          222 
## -0.236510673 -0.011232245  0.630867557  0.965698775  0.842951821  0.980329919 
##          223          224          225          226          227          228 
##  0.405603758 -0.081259496  0.020476796  0.198995617  0.539530739  0.695241902 
##          229          230          231          232          233          234 
##  0.505353089  0.103543071  0.084199783 -0.060773234 -0.687262335  0.223897168 
##          235          236          237          238          239          240 
##  0.158265282 -0.097488959 -0.063130119 -0.320686525 -0.111602996  0.223043923 
##          241          242          243          244          245          246 
##  0.069286926 -0.075827417 -0.195341651  0.007382603 -0.155997697 -0.340909232 
##          247          248          249          250          251          252 
## -0.313957509 -0.278744127 -0.331574026 -0.279316587 -0.669261650 -0.511348060 
##          253          254          255          256          257          258 
## -0.543592370 -0.567951602 -0.649225670 -0.795434355 -1.068118107 -0.536898969 
##          259          260          261          262          263          264 
## -0.781653867 -1.378215169 -0.642796987 -0.695479334 -1.133816241 -0.705216939 
##          265          266          267          268          269          270 
## -0.833183745 -0.481082101 -0.949364510 -0.666934935 -0.729533596 -0.670244008 
##          271          272          273          274          275          276 
## -0.459775767 -0.312795933 -0.683795518 -0.958265199 -0.710554591 -0.461965690 
##          277          278          279          280          281          282 
## -1.927703353 -1.860354405 -1.788623255 -0.580177525 -1.364166882 -1.264233742 
##          283          284          285          286          287          288 
## -1.405559426 -0.994684914 -1.846049333 -1.169577240 -1.522802689 -1.370957154 
##          289          290          291          292          293          294 
## -1.488322029 -0.898873445 -1.182528558 -1.869183395 -1.303331789 -0.838020471 
##          295          296          297          298          299          300 
## -0.783640584 -0.184541374 -0.804762372 -0.845862677 -0.639393492 -0.073034083 
##          301          302          303          304          305          306 
##  0.505106704 -0.414760325 -0.152886163  0.011629339  0.175510337  0.008692918 
##          307          308          309          310          311          312 
## -0.466941362 -0.303730275  0.212979426 -0.354538102 -0.630846915 -0.610448748 
##          313          314          315          316          317          318 
## -1.751952004 -1.272773337 -0.374306882  0.290963727  0.046446696 -0.214981661 
##          319          320          321          322          323          324 
## -0.545341288 -0.332527264 -0.469964243 -0.455682589 -0.018678859 -0.219464918 
##          325          326          327          328          329          330 
## -0.255516321  0.299562431  0.462774125 -0.712331746  0.204310949  0.378700005 
##          331          332          333          334          335          336 
##  0.236595198  0.131521713  0.023084524  0.309143039  0.393019139  0.514684828 
##          337          338          339          340          341          342 
##  0.709765643  0.815688120  0.422194522  0.927938263  1.026089235  0.119582660 
##          343          344          345          346          347          348 
## -0.400948873  0.618876707  1.283429018  1.428341833  1.416771338  1.546686323 
##          349          350          351          352          353          354 
##  0.844014152  0.520292465  0.964667125  1.468735348  1.440001829  1.088767430 
##          355          356          357          358          359          360 
##  1.381710323  1.285408988  0.927875505  0.875859442  1.146208669  1.053124920 
##          361          362          363          364          365          366 
##  0.992434881  1.103931611  1.235399280  0.792817833  0.536064069  1.022193226 
##          367          368          369          370          371          372 
##  1.369988900  1.386744406  1.145528674  0.111164949 -0.122587151  0.403142012 
##          373          374          375          376          377          378 
##  0.917741437  0.574358546  0.414126633  0.390922055  0.409998132 -0.047921889 
##          379          380          381          382          383          384 
##  0.051743407  0.346350681  0.219834318  0.183649959  0.003918620 -0.087689089 
##          385          386          387          388          389          390 
## -0.443879508 -0.368255592 -0.233817769 -0.134835441 -0.394255333 -0.319883083 
##          391          392          393          394          395          396 
## -0.218698343 -0.910699733 -0.465972929 -0.585333047 -0.694596814 -0.586108231 
##          397          398          399          400          401          402 
## -0.738585332 -0.679538577 -1.118610524 -0.420766359 -0.638185474 -0.714555809 
##          403          404          405          406          407          408 
## -0.585440652 -0.563505308 -0.878102761 -1.280603695 -1.191681077 -0.934102963 
##          409          410          411          412          413          414 
## -1.596363594 -1.557809531 -0.722601433 -1.255975639 -0.770097847 -0.727531531 
##          415          416          417          418          419          420 
## -0.518165467 -1.293445438 -0.778215674 -0.565879295 -0.856861437 -0.400860881 
##          421          422          423          424          425          426 
## -0.440988575 -0.959018978 -0.689239873 -0.689466316 -0.532029981 -0.278728684 
##          427          428          429          430          431          432 
##  0.266328315  0.783612944 -0.362063857 -0.402825267 -0.403869896 -0.366283882 
##          433          434          435          436          437          438 
## -0.063202851  0.165020700  0.151977809 -0.472627095 -0.542580764 -0.820343633 
##          439          440          441          442          443          444 
## -0.583075855 -0.521115134 -0.403670372 -0.526089683 -0.566016826 -1.104702309 
##          445          446          447          448          449          450 
## -0.806441249 -0.082094347 -0.979519688 -0.878787309 -0.900340594 -0.696046618 
##          451          452          453          454          455          456 
## -0.459637216  0.026626468 -0.546385035 -0.882709570 -1.394742797 -1.432171235 
##          457          458          459          460          461          462 
## -0.853846173 -0.715528872 -1.211671395 -1.384127172 -1.836427085 -0.407158641 
##          463          464          465          466          467          468 
## -1.607260055 -0.880169779 -0.844179745 -0.666642593 -0.660131954 -0.278294858 
##          469          470          471          472          473          474 
## -1.266666170 -1.041434986 -0.633569678 -0.426953761 -0.315162411 -0.559516785 
##          475          476 
## -0.196315618 -0.496435946

exp(f_dl14_hanover$forecasts[1,2])
## [1] 2.365621
exp(f_dl14_hanover$forecasts[1,1])
## [1] 0.5306325
exp(f_dl14_hanover$forecasts[1,3])
## [1] 9.101989
exp(f_dl14_hanover$forecasts[1,2]) - exp(full_cases_wastewater_weather_data_hanover_test[1,6])
## [1] 1.257414
exp(f_dl14_hanover$forecasts[7,2])
## [1] 2.88277
exp(f_dl14_hanover$forecasts[7,1])
## [1] 0.6819826
exp(f_dl14_hanover$forecasts[7,3])
## [1] 11.4008
exp(f_dl14_hanover$forecasts[7,2]) - exp(full_cases_wastewater_weather_data_hanover_test[7,6])
## [1] 0.6243629
exp(f_dl14_hanover$forecasts[14,2])
## [1] 3.742474
exp(f_dl14_hanover$forecasts[14,1])
## [1] 0.8179684
exp(f_dl14_hanover$forecasts[14,3])
## [1] 15.20064
exp(f_dl14_hanover$forecasts[14,2]) - exp(full_cases_wastewater_weather_data_hanover_test[14,6])
## [1] 2.903247
mod_dl2_hanover <- dlm(log_mean_new_cases ~ log_viral_gene,
                       data = full_cases_wastewater_weather_data_hanover_train,
                       q=2)
summary(mod_dl2_hanover)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.00543 -0.54939  0.01187  0.60816  1.80704 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -7.336034   0.316832 -23.154  < 2e-16 ***
## log_viral_gene.t  0.254708   0.060181   4.232 2.77e-05 ***
## log_viral_gene.1  0.006965   0.083828   0.083    0.934    
## log_viral_gene.2  0.266634   0.060184   4.430 1.16e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7937 on 484 degrees of freedom
## Multiple R-squared:  0.5591, Adjusted R-squared:  0.5564 
## F-statistic: 204.6 on 3 and 484 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 1165.335 1186.286
f_dl2_hanover <- forecast(mod_dl2_hanover,
                          x= t(full_cases_wastewater_weather_data_hanover_test[,7]),
                          h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_dl2_hanover$forecasts) 
## [1] 0.6313079
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_dl2_hanover$forecasts) 
## [1] 0.5276261
checkresiduals(mod_dl2_hanover)
##            1            2            3            4            5            6 
##  1.074904378  0.909580304  0.813942755  0.847367928  0.915176778  0.825297521 
##            7            8            9           10           11           12 
##  0.512100817  0.364578861  0.554887522  0.800847849  0.692634327  0.811843633 
##           13           14           15           16           17           18 
##  0.754998314  0.319176751  0.415253152  0.485187186  0.685511485  0.769046660 
##           19           20           21           22           23           24 
##  0.853513255  0.886242227  0.591169007  0.530373813  0.388556736  0.239233951 
##           25           26           27           28           29           30 
##  1.212386581  0.640055757  1.185694520  0.439720833  0.907900024  0.759105273 
##           31           32           33           34           35           36 
##  0.768706013  1.110752752  1.070964950  1.043351598  0.711022728  0.577947075 
##           37           38           39           40           41           42 
##  0.849282961  0.826891381  1.019611241  0.491621656  0.187619626  0.513869464 
##           43           44           45           46           47           48 
##  0.363113094  0.301199723  0.036692991  0.303271139 -0.194842763  0.233089707 
##           49           50           51           52           53           54 
##  0.224898882 -0.102113259  0.633930112  0.309632108  0.587484382  0.731570072 
##           55           56           57           58           59           60 
##  0.790867520  0.533382024  0.367530334  0.521603643  0.977218594  1.123166756 
##           61           62           63           64           65           66 
##  1.684570018  1.372199738  0.837337619  1.705404861  1.509916382  1.168745119 
##           67           68           69           70           71           72 
##  0.692708975  0.721212296  0.450098132  0.362010611  0.571254503  0.454525067 
##           73           74           75           76           77           78 
##  0.654534606  0.956698153  0.924937603  0.802896314  0.982634576  1.378262422 
##           79           80           81           82           83           84 
##  1.807038228  1.581280092  1.395405579  0.992030019  1.216424096  1.047661757 
##           85           86           87           88           89           90 
##  1.278679605  0.726198923  0.601657287  0.344861700  0.174968971 -0.150744724 
##           91           92           93           94           95           96 
##  0.314674818  0.068024020  0.825029557  0.716439430  0.346274113  0.549324351 
##           97           98           99          100          101          102 
##  0.667736988  0.841428067  0.690830431  0.706628165  0.640008480  0.857655665 
##          103          104          105          106          107          108 
##  0.443817988 -0.620852423  0.782301579  0.400309839  0.862209606  0.540264479 
##          109          110          111          112          113          114 
##  0.935535765  1.098149659 -0.448055580  1.045665826  0.884899448  0.992795000 
##          115          116          117          118          119          120 
##  0.725535486  0.863861034 -0.015592757  0.421754495  0.643680411  0.121095446 
##          121          122          123          124          125          126 
##  0.192444848 -0.428640652  0.587991404  1.317571670  0.630410289  1.030016778 
##          127          128          129          130          131          132 
##  0.638061548  0.853042507  0.175554177  0.408035589 -0.385823644 -0.341518980 
##          133          134          135          136          137          138 
##  0.064563721 -0.873256926 -0.863441094  0.336990101  0.551495098  0.164858545 
##          139          140          141          142          143          144 
##  0.144624688 -0.614578540  0.671125231 -0.756484243  0.329425517 -0.214051631 
##          145          146          147          148          149          150 
##  0.502312457  0.546617121 -0.069881095  0.038038256  0.010827720 -0.382068463 
##          151          152          153          154          155          156 
## -0.288709572 -0.430281939 -0.385977275 -0.658769107 -0.541451645 -0.208861649 
##          157          158          159          160          161          162 
##  0.009898374  0.053678818  0.545467696  0.589772360 -0.482069286 -0.324318278 
##          163          164          165          166          167          168 
## -0.825355062 -0.263190827 -0.211678219  0.454882239  0.434648382 -0.199400825 
##          169          170          171          172          173          174 
## -0.027422857 -0.073006213 -0.010906411 -0.631366195  0.403108800 -0.898594398 
##          175          176          177          178          179          180 
## -1.520465388 -1.488680646 -1.585875608 -0.841148672 -0.733545036 -0.376191059 
##          181          182          183          184          185          186 
##  0.534298671 -0.087572319  0.164206090 -0.021450911 -1.104487756 -1.484003685 
##          187          188          189          190          191          192 
## -1.160780251 -0.691958622 -1.584119942 -1.844298293 -0.319388955 -0.501017433 
##          193          194          195          196          197          198 
## -0.075298339 -0.695566906 -1.264062160 -1.885933150 -0.803682692 -0.696257483 
##          199          200          201          202          203          204 
## -0.338376949 -0.408069614 -0.001145133  0.399633113  0.045903040  0.031020060 
##          205          206          207          208          209          210 
## -0.058575103  0.091698561 -0.022846332  0.020627883  0.269372440  0.138087858 
##          211          212          213          214          215          216 
## -0.073645921 -0.171275771 -0.351099196  0.136452850 -0.026460611  0.108626959 
##          217          218          219          220          221          222 
##  0.102108428  0.116281506  0.009686828 -0.280233188  0.048205423 -0.098274364 
##          223          224          225          226          227          228 
##  0.292856395  0.344645698 -0.090399456 -0.031073738 -0.055136671 -0.062179566 
##          229          230          231          232          233          234 
## -0.246194888  0.252092698  0.850831660  1.237670317  1.143743181  1.619840026 
##          235          236          237          238          239          240 
##  0.767348523  0.195591282 -0.231748279  0.005819147  0.745751734  0.687074491 
##          241          242          243          244          245          246 
##  1.359181503  0.605556514  0.413252689 -0.464306089 -0.960536780  0.294457612 
##          247          248          249          250          251          252 
##  0.012906746 -0.025883720  0.129434422 -0.224335611  0.042170231  0.406827927 
##          253          254          255          256          257          258 
##  0.270410106  0.117429426 -0.063906977  0.279615235  0.017803048  0.108054453 
##          259          260          261          262          263          264 
##  0.074863383 -0.049892035 -0.120705750 -0.072631289 -0.585800727 -0.469975543 
##          265          266          267          268          269          270 
## -0.720280463 -0.711112658 -0.728718820 -0.901676184 -1.050820970 -0.510362668 
##          271          272          273          274          275          276 
## -0.713126053 -1.280942318 -0.549380611 -0.567964221 -1.032067459 -0.655967313 
##          277          278          279          280          281          282 
## -0.623423051 -0.343770228 -0.427801767 -0.221350798 -0.464952907 -0.413393550 
##          283          284          285          286          287          288 
## -0.248467117 -0.147322908 -0.609797569 -0.874112889 -0.643772004 -0.444892033 
##          289          290          291          292          293          294 
## -1.944570370 -1.745399112 -1.749299537 -0.550808050 -1.418777491 -1.309182436 
##          295          296          297          298          299          300 
## -1.383076828 -1.022143134 -1.784001214 -1.131698130 -1.515964517 -1.451359863 
##          301          302          303          304          305          306 
## -1.549982710 -0.906686639 -1.223174104 -1.843346374 -0.979941375 -0.622732221 
##          307          308          309          310          311          312 
##  0.282213021  0.705465634 -0.231885528 -0.329258941 -0.138082096  0.130845738 
##          313          314          315          316          317          318 
##  0.462300073 -0.984404811 -0.647165072 -0.535054591 -0.405729946 -0.569523190 
##          319          320          321          322          323          324 
## -1.136348496 -0.698704457 -0.519517315 -0.984972619 -0.913278314 -0.842496058 
##          325          326          327          328          329          330 
## -1.931486426 -1.366976761 -0.396259062  0.210458594 -0.024744834 -0.208430383 
##          331          332          333          334          335          336 
## -0.563807295 -0.372795437 -0.475574344 -0.468552160 -0.035811463 -0.230524520 
##          337          338          339          340          341          342 
##  0.103203702  0.545222115  1.790948541  0.015468148  0.630274318 -0.378547375 
##          343          344          345          346          347          348 
## -0.360453450 -0.111905729 -0.549410048 -0.046702037  0.081118036  0.247193439 
##          349          350          351          352          353          354 
##  0.209422320  0.448524617  0.295296776  0.870686488  0.855363700  0.093149545 
##          355          356          357          358          359          360 
## -0.392268395  0.615982736  1.282396486  1.380177482  1.391941158  1.332181455 
##          361          362          363          364          365          366 
##  0.670250267  0.447404596  0.874816000  1.394506394  1.333724322  1.056803484 
##          367          368          369          370          371          372 
##  1.100546218  1.080636225  0.878085006  0.855951883  1.129774860  1.070554033 
##          373          374          375          376          377          378 
##  1.074192757  1.115652791  1.317805979  0.942873396  0.819657014  1.271029277 
##          379          380          381          382          383          384 
##  1.654219973  1.647530803  1.562742180  0.385583014  0.090231408  0.319106254 
##          385          386          387          388          389          390 
##  0.876061566  0.634780692  0.405024148  0.506976088  0.625122014  0.159180842 
##          391          392          393          394          395          396 
##  0.503795194  0.759402681  0.622038863  0.549350712  0.432401384  0.304195068 
##          397          398          399          400          401          402 
## -0.199656419 -0.166990674 -0.040097793  0.007171815 -0.311540737 -0.126567776 
##          403          404          405          406          407          408 
##  0.034336104 -0.713011779  0.009769598 -0.166971027 -0.310266644 -0.262247702 
##          409          410          411          412          413          414 
## -0.230346777 -0.250946422 -0.881167195 -0.219000473 -0.453105932 -0.567549918 
##          415          416          417          418          419          420 
## -0.541585696 -0.316158251 -0.644442267 -1.157060992 -0.944130870 -0.714249650 
##          421          422          423          424          425          426 
## -1.410037624 -1.441554786 -0.503370230 -1.015603107 -0.676855271 -0.478689460 
##          427          428          429          430          431          432 
## -0.309432861 -1.279358741 -0.795123629 -0.774958278 -0.870051628 -0.460562987 
##          433          434          435          436          437          438 
##  0.064269616 -0.557601374 -0.527957560 -0.496625530 -0.384213673 -0.086650305 
##          439          440          441          442          443          444 
##  0.315711420  1.196303375 -0.054176275 -0.558966810 -0.547926651 -1.123159358 
##          445          446          447          448          449          450 
## -0.633630321 -0.288301903  0.161797143 -0.520698469 -0.970158721 -1.018230265 
##          451          452          453          454          455          456 
## -1.474917212 -1.241433145 -0.840823567 -0.655920105 -0.754542951 -1.509717883 
##          457          458          459          460          461          462 
## -1.019416515 -0.787322851 -1.481948875 -1.136738003 -0.691139116 -0.581544061 
##          463          464          465          466          467          468 
## -0.620179678  0.013573898 -1.116517831 -1.380717737 -1.685965964 -1.641661300 
##          469          470          471          472          473          474 
## -1.080081121 -0.962784799 -1.336842529 -1.538708273 -2.005426655 -0.421991549 
##          475          476          477          478          479          480 
## -1.609992393 -0.910107543 -0.866649761 -0.693356531 -0.848838245 -0.388398991 
##          481          482          483          484          485          486 
## -1.329900388 -0.994179727 -0.595860390 -0.305015606 -0.186189695 -0.137481165 
##          487          488 
##  0.072232710 -0.307177484

mod_dl13_hanover <- dlm(log_mean_new_cases ~ log_viral_gene,
                       data = full_cases_wastewater_weather_data_hanover_train,
                       q=13)
summary(mod_dl13_hanover)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9269 -0.5394  0.0253  0.5566  1.5463 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -8.266744   0.321639 -25.702   <2e-16 ***
## log_viral_gene.t   0.162195   0.059105   2.744   0.0063 ** 
## log_viral_gene.1   0.036405   0.080869   0.450   0.6528    
## log_viral_gene.2  -0.009323   0.080998  -0.115   0.9084    
## log_viral_gene.3   0.057573   0.082325   0.699   0.4847    
## log_viral_gene.4   0.108674   0.082369   1.319   0.1877    
## log_viral_gene.5   0.010806   0.083125   0.130   0.8966    
## log_viral_gene.6   0.021985   0.083091   0.265   0.7914    
## log_viral_gene.7   0.019192   0.083111   0.231   0.8175    
## log_viral_gene.8   0.090591   0.083330   1.087   0.2775    
## log_viral_gene.9  -0.044655   0.082663  -0.540   0.5893    
## log_viral_gene.10  0.021492   0.082655   0.260   0.7950    
## log_viral_gene.11  0.083382   0.081444   1.024   0.3065    
## log_viral_gene.12  0.015224   0.081440   0.187   0.8518    
## log_viral_gene.13  0.016637   0.059523   0.280   0.7800    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7452 on 462 degrees of freedom
## Multiple R-squared:  0.6203, Adjusted R-squared:  0.6088 
## F-statistic: 53.91 on 14 and 462 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 1089.883 1156.563
f_dl13_hanover <- forecast(mod_dl13_hanover,
                          x= t(full_cases_wastewater_weather_data_hanover_test[,7]),
                          h=14)
rmse(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
     f_dl13_hanover$forecasts) 
## [1] 0.5809196
mae(full_cases_wastewater_weather_data_hanover_test$log_mean_new_cases,
    f_dl13_hanover$forecasts) 
## [1] 0.4512274
checkresiduals(mod_dl13_hanover)
##            1            2            3            4            5            6 
##  0.774788470  0.744986890  0.310267632  0.377049317  0.472122652  0.615842951 
##            7            8            9           10           11           12 
##  0.711424358  0.783185854  0.848608783  0.589622348  0.492853798  0.450875717 
##           13           14           15           16           17           18 
##  0.240777065  1.122294650  0.582931274  0.909985345  0.218732044  0.690836860 
##           19           20           21           22           23           24 
##  0.602675907  0.523796450  0.874284496  1.161813927  1.082547651  0.659340026 
##           25           26           27           28           29           30 
##  0.585210885  0.784155229  0.778580745  0.962826231  0.682649131  0.374406556 
##           31           32           33           34           35           36 
##  0.589759882  0.439354491  0.334767929  0.026509367  0.292449497 -0.154979209 
##           37           38           39           40           41           42 
##  0.279393601  0.125351453 -0.192808619  0.247397637 -0.045946718  0.366809223 
##           43           44           45           46           47           48 
##  0.577278030  0.679749794  0.447464955  0.353164730  0.523188470  0.728504280 
##           49           50           51           52           53           54 
##  1.018313373  0.968222238  0.830446865  0.566907429  1.406217062  1.377208508 
##           55           56           57           58           59           60 
##  1.187276230  0.830754896  1.284826430  0.972990986  0.845031305  1.020313999 
##           61           62           63           64           65           66 
##  0.865147233  1.007257915  1.145776343  1.261468134  1.116165987  0.988965763 
##           67           68           69           70           71           72 
##  1.392313541  1.393580356  1.286913304  1.250360786  1.127511846  1.362147560 
##           73           74           75           76           77           78 
##  1.114204166  1.436643633  0.823277829  0.803704180  0.555877495  0.872252402 
##           79           80           81           82           83           84 
##  0.493581707  0.619129675  0.347525367  0.763454761  0.638586080  0.270055232 
##           85           86           87           88           89           90 
##  0.546519601  0.689464465  0.769864796  0.672985955  0.638360305  0.546502662 
##           91           92           93           94           95           96 
##  0.877918177  0.480613375 -0.533594559  0.837080099  0.441750729  0.798189632 
##           97           98           99          100          101          102 
##  0.471521585  0.927905155  1.150551679 -0.365891073  1.095341161  0.957338871 
##          103          104          105          106          107          108 
##  0.977303104  0.726425984  0.923026454  0.188857256  0.641405764  0.832204328 
##          109          110          111          112          113          114 
##  0.296711253  0.423878760 -0.473482685  0.588434857  0.709145143  0.175470993 
##          115          116          117          118          119          120 
##  0.865882057  0.421420407  0.936374177  0.301665362  0.620018548  0.011011473 
##          121          122          123          124          125          126 
##  0.051723845  0.531600047 -0.485780831 -0.380418816  0.628788477  0.716913612 
##          127          128          129          130          131          132 
##  0.071841449  0.105413905 -0.546628972  0.676526049 -0.497096788  0.383093497 
##          133          134          135          136          137          138 
## -0.100305661  0.176308887  0.335366733 -0.049110705 -0.009301489  0.068129770 
##          139          140          141          142          143          144 
## -0.195381071 -0.079586626  0.210419940  0.196961696 -0.275896594 -0.143128130 
##          145          146          147          148          149          150 
## -0.180003267  0.002847022  0.055304027  0.556564247  0.640825883 -0.371840798 
##          151          152          153          154          155          156 
## -0.187518832 -0.324557289  0.002303091  0.087227432  0.350182006  0.438701013 
##          157          158          159          160          161          162 
## -0.029045593 -0.004834965  0.094553008  0.101771040 -0.268501898  0.614349187 
##          163          164          165          166          167          168 
##  0.061767300 -0.586762593 -0.798908356 -0.888205260 -0.101138684 -0.369639854 
##          169          170          171          172          173          174 
## -0.196634633 -0.046150393 -0.547514341  0.014461102 -0.266992717 -0.708934597 
##          175          176          177          178          179          180 
## -1.008430549 -0.720543527  0.053047615 -0.922935279 -1.247687905  0.151235684 
##          181          182          183          184          185          186 
## -0.202870074  0.233058608 -0.627941961 -1.028431693 -1.722841186 -0.707015985 
##          187          188          189          190          191          192 
## -0.560779092  0.032993665 -0.148814259  0.194153377  0.459734866  0.123055279 
##          193          194          195          196          197          198 
##  0.128923127 -0.061869293  0.206754363  0.081914160  0.045076967  0.318980533 
##          199          200          201          202          203          204 
##  0.162548758  0.039540998 -0.086887354 -0.018984925  0.372282812  0.081729720 
##          205          206          207          208          209          210 
##  0.124475531  0.110667272  0.157256726 -0.028630634 -0.160968272  0.095037633 
##          211          212          213          214          215          216 
## -0.132578852  0.139858254  0.194872707 -0.097449007 -0.087710852  0.200711310 
##          217          218          219          220          221          222 
##  0.042721744 -0.236465873 -0.014152501  0.614388266  0.963617285  0.845120391 
##          223          224          225          226          227          228 
##  0.987447648  0.399712347 -0.082579022  0.025301040  0.178778739  0.541528949 
##          229          230          231          232          233          234 
##  0.695648070  0.531946918  0.113037886  0.079329676 -0.068881612 -0.647177877 
##          235          236          237          238          239          240 
##  0.227123054  0.151337728 -0.157121913 -0.046052246 -0.323325870 -0.122273943 
##          241          242          243          244          245          246 
##  0.290843118  0.068676742 -0.079275514 -0.257593745  0.010125406 -0.154942970 
##          247          248          249          250          251          252 
## -0.337275150 -0.301100832 -0.279527459 -0.329204608 -0.253721288 -0.669781303 
##          253          254          255          256          257          258 
## -0.512419296 -0.540599736 -0.567146843 -0.650645701 -0.797176479 -1.047242222 
##          259          260          261          262          263          264 
## -0.539414751 -0.783869264 -1.386394202 -0.648836280 -0.696045039 -1.136111557 
##          265          266          267          268          269          270 
## -0.732849422 -0.830877595 -0.480399593 -0.948837092 -0.654954675 -0.730548209 
##          271          272          273          274          275          276 
## -0.667938010 -0.446138390 -0.313021373 -0.683641094 -0.952095913 -0.707798809 
##          277          278          279          280          281          282 
## -0.463436606 -1.926916936 -1.821256891 -1.790765283 -0.582371330 -1.367892111 
##          283          284          285          286          287          288 
## -1.267663332 -1.406530741 -0.996849874 -1.853436514 -1.168182165 -1.525007463 
##          289          290          291          292          293          294 
## -1.374388115 -1.478453920 -0.900224229 -1.185043816 -1.878780313 -1.302689498 
##          295          296          297          298          299          300 
## -0.836590365 -0.785759143 -0.176223174 -0.804897533 -0.839923028 -0.630459620 
##          301          302          303          304          305          306 
## -0.067476028  0.504647131 -0.396727934 -0.126155303  0.008627863  0.172520522 
##          307          308          309          310          311          312 
##  0.091571689 -0.474259249 -0.311014011  0.190630195 -0.375910125 -0.634555994 
##          313          314          315          316          317          318 
## -0.620097844 -1.838607224 -1.278530598 -0.377264760  0.282523383  0.029353404 
##          319          320          321          322          323          324 
## -0.216191038 -0.548643509 -0.357213877 -0.469242105 -0.457389049 -0.021692691 
##          325          326          327          328          329          330 
## -0.212770581 -0.256569655  0.301780910  0.459724999 -0.710959005  0.199726781 
##          331          332          333          334          335          336 
##  0.383602305  0.248337722  0.135864346  0.015516911  0.311666401  0.414498575 
##          337          338          339          340          341          342 
##  0.507365101  0.684646531  0.902496977  0.419097459  0.917110623  0.891755853 
##          343          344          345          346          347          348 
##  0.116473068 -0.402330123  0.616103534  1.274606872  1.426607373  1.414402646 
##          349          350          351          352          353          354 
##  1.546341994  0.840501804  0.518584663  0.959608108  1.457521203  1.437402473 
##          355          356          357          358          359          360 
##  1.085428745  1.372449000  1.278043644  0.926514617  0.869893804  1.119305861 
##          361          362          363          364          365          366 
##  1.050880532  0.990437398  1.095378230  1.228622730  0.792698002  0.533158658 
##          367          368          369          370          371          372 
##  0.997300573  1.369735844  1.387962651  1.148842028  0.111342116 -0.123421763 
##          373          374          375          376          377          378 
##  0.405276927  0.919350649  0.574777456  0.412535968  0.401749220  0.412713982 
##          379          380          381          382          383          384 
## -0.048624621  0.044425483  0.358254406  0.220227677  0.184274065 -0.024578248 
##          385          386          387          388          389          390 
## -0.083486178 -0.443006462 -0.360932533 -0.216488431 -0.134808479 -0.393002109 
##          391          392          393          394          395          396 
## -0.284912955 -0.217152201 -0.910757658 -0.468644185 -0.576666118 -0.694444883 
##          397          398          399          400          401          402 
## -0.583986570 -0.742317313 -0.675351742 -1.118187567 -0.413250045 -0.618778857 
##          403          404          405          406          407          408 
## -0.714311475 -0.583969108 -0.527272818 -0.873728486 -1.281744832 -1.193543856 
##          409          410          411          412          413          414 
## -0.912483256 -1.596773430 -1.557833890 -0.726948228 -1.251785632 -0.770482170 
##          415          416          417          418          419          420 
## -0.724623437 -0.496638591 -1.294718536 -0.778456499 -0.546260259 -0.855888835 
##          421          422          423          424          425          426 
## -0.400622518 -0.441294023 -0.949167083 -0.692229365 -0.686973035 -0.513527566 
##          427          428          429          430          431          432 
## -0.283084888  0.267373262  0.792611920 -0.371144364 -0.405178932 -0.402864782 
##          433          434          435          436          437          438 
## -0.303682514 -0.066089440  0.163773350  0.149197364 -0.476710102 -0.549930349 
##          439          440          441          442          443          444 
## -0.823372487 -0.566266857 -0.541877202 -0.405083165 -0.529610829 -0.639083914 
##          445          446          447          448          449          450 
## -1.111222926 -0.808917389 -0.051019902 -0.998746713 -0.879704351 -0.906377555 
##          451          452          453          454          455          456 
## -0.763763607 -0.464301498  0.025522948 -0.530598354 -0.895526523 -1.395853208 
##          457          458          459          460          461          462 
## -1.432745566 -0.899089993 -0.720543601 -1.212967195 -1.353783950 -1.852105881 
##          463          464          465          466          467          468 
## -0.408947493 -1.613523935 -0.935683923 -0.845425136 -0.667414353 -0.663817552 
##          469          470          471          472          473          474 
## -0.279279232 -1.268011300 -1.045359280 -0.634008213 -0.429224601 -0.315341873 
##          475          476          477 
## -0.563285865 -0.200609987 -0.498041216

wake_mod_dl14_res <- ggAcf(residuals(mod_dl14)) + 
  theme_bw(base_size = 15) + ggtitle("")
##            1            2            3            4            5            6 
## -0.151770801 -0.160174319 -0.006891305  0.012071577 -0.068008232  0.122592200 
##            7            8            9           10           11           12 
##  0.015847908 -0.178071430  0.214099747 -0.006353717 -0.012215233 -0.298606308 
##           13           14           15           16           17           18 
##  0.354637808  0.025646543 -0.260584528  0.180020051  0.013853802  0.158328818 
##           19           20           21           22           23           24 
##  0.278353673  0.210017083 -0.078877295 -0.348110739  0.055222675 -0.110048519 
##           25           26           27           28           29           30 
## -0.061257221 -0.044888949 -0.037927120 -0.391780768 -0.357219577 -0.325654025 
##           31           32           33           34           35           36 
## -0.173423164 -0.161965045 -0.331116892 -0.070850976 -0.327220624 -0.077795680 
##           37           38           39           40           41           42 
## -0.192371480 -0.199630447 -0.499721719 -0.339997397 -0.186092416 -0.112659390 
##           43           44           45           46           47           48 
##  0.046153616 -0.065795631 -0.027007380  0.272730134  0.385322871  0.284586325 
##           49           50           51           52           53           54 
##  0.408328451  0.418240641  0.607870058  0.227049609  0.244602361  0.437649300 
##           55           56           57           58           59           60 
##  0.361156912  0.546368097  0.755777260  0.480700608  0.637950909  0.761534885 
##           61           62           63           64           65           66 
##  0.367490147  0.299815161  0.125096013  0.327222264  0.264879971  0.021003598 
##           67           68           69           70           71           72 
## -0.314816009 -0.158956628 -0.277862458 -0.061712910  0.215764953 -0.300915457 
##           73           74           75           76           77           78 
##  0.034125718  0.320742606 -0.013196360 -0.464188949 -0.230510455  0.398974451 
##           79           80           81           82           83           84 
##  0.678452605  0.333376179  0.221424191  0.490438548  0.447209701  0.534268079 
##           85           86           87           88           89           90 
##  0.709815225  0.119640883  0.504932076  0.261929540  0.246888146  0.194527175 
##           91           92           93           94           95           96 
##  0.178860328  0.206730858  0.252978892  0.005364263  0.074426266  0.216287910 
##           97           98           99          100          101          102 
##  0.232817539  0.472290377  0.614895530  0.447324577  0.424654756  0.445229243 
##          103          104          105          106          107          108 
##  0.310757531  0.385827965  0.093229086  0.539775588  0.138891230  0.432588068 
##          109          110          111          112          113          114 
##  0.280276822  0.381751416  0.606126225  0.676981509  0.861906417  0.755161556 
##          115          116          117          118          119          120 
##  0.905947690  1.096363380  0.879085200  0.986306869  0.824226121  0.934572457 
##          121          122          123          124          125          126 
##  0.910965863  1.303534614  0.903807919  0.996189016  0.642240600  0.893461200 
##          127          128          129          130          131          132 
##  1.098934200  0.717405303  0.966141121  0.347449470  0.226289563  0.190708147 
##          133          134          135          136          137          138 
##  0.279640583  0.546293239 -0.972154654 -0.343724632 -1.439011674 -0.468140737 
##          139          140          141          142          143          144 
## -0.038232381  0.061734132  0.289038977 -0.648272094 -0.593070505 -0.507174123 
##          145          146          147          148          149          150 
## -0.433670957 -0.830026838 -0.441469781 -0.549782252 -0.114253460 -0.211018737 
##          151          152          153          154          155          156 
## -0.002542970 -0.025310874  0.152103161  0.021355744 -0.153498703 -0.488846098 
##          157          158          159          160          161          162 
## -0.292301718 -0.917386352 -0.565715895 -0.853973793 -0.323561354 -0.431294276 
##          163          164          165          166          167          168 
## -0.624386866 -0.973825825 -0.870499160 -0.656768083 -0.559515570 -0.817463012 
##          169          170          171          172          173          174 
## -2.439120965 -1.011462182 -0.885873005 -0.813675933 -0.818385289 -0.226150827 
##          175          176          177          178          179          180 
## -0.307792930 -0.183935955 -0.076929703 -0.215211957 -0.557805408  0.148763847 
##          181          182          183          184          185          186 
## -0.101680887 -0.117617739  0.143736499  0.020394433  0.337207754  0.045956078 
##          187          188          189          190          191          192 
##  0.332593271  0.256071911  0.247573241  0.629550849  0.509287247  0.192702377 
##          193          194          195          196          197          198 
##  0.071292443  0.565404648  0.436179259  0.301658228  0.232234299  0.393091647 
##          199          200          201          202          203          204 
##  0.079781736  0.118662310  0.338854035  0.045047977  0.088193667  0.237000094 
##          205          206          207          208          209          210 
##  0.238516762  0.086160984  0.041897754  0.315808665  0.229016112  0.125181519 
##          211          212          213          214          215          216 
##  0.151657938  0.337117793  0.162295906  0.296032161  0.514344361  0.468765657 
##          217          218          219          220          221          222 
##  0.496941398  0.346308667  0.712701681  0.474850754  0.594063627  0.649302975 
##          223          224          225          226          227          228 
##  0.655152141  1.093468567  0.364177085  0.531892550  0.388969558  0.627855968 
##          229          230          231          232          233          234 
##  0.558245460  0.300468018  0.212893475 -0.522911391 -1.841610748 -0.155285556 
##          235          236          237          238          239          240 
## -0.508681021 -0.579301985 -0.268928725 -0.189132236 -0.548010627 -0.388553086 
##          241          242          243          244          245          246 
## -0.570005532 -0.318321249 -0.476077039 -0.551777154 -0.429298466 -0.723291715 
##          247          248          249          250          251          252 
## -0.497671782 -0.548269407 -0.597052392 -0.719545452 -0.418849460 -0.672944848 
##          253          254          255          256          257          258 
## -1.010622705 -0.327623496 -0.575844083 -0.556350035 -0.523688877 -0.519657389 
##          259          260          261          262          263          264 
## -0.268375536 -0.803853303 -0.610959894 -0.710925756 -0.510139753 -0.539475414 
##          265          266          267          268          269          270 
##  0.234843138 -0.608050746 -0.027640151 -0.374300241 -0.212675738  0.334159885 
##          271          272          273          274          275          276 
##  0.076729409 -0.369232372 -0.523525111 -1.050303551 -0.205229336 -0.283065176 
##          277          278          279          280          281          282 
## -0.125127059 -2.606668860 -1.887651121  0.137770559 -0.614603483 -0.393221520 
##          283          284          285          286          287          288 
## -0.027077421 -0.031360472 -0.227814319 -0.047417728 -0.020386512 -0.539459137 
##          289          290          291          292          293          294 
## -0.383600678 -0.422027824 -0.323895183 -0.631557439 -0.365336871 -0.369111965 
##          295          296          297          298          299          300 
## -0.035236118 -0.547808614 -1.315971321 -0.329583472 -0.750814942 -0.091505927 
##          301          302          303          304          305          306 
## -0.482518346 -0.259376639 -0.122673196 -0.359898283  0.033848038  0.335394645 
##          307          308          309          310          311          312 
##  0.524354738  0.927982380  0.780829953  0.418791558  0.186413545  0.182083536 
##          313          314          315          316          317          318 
## -0.373833890  0.127315760  0.390806471  0.138672724  0.573963506  0.458075879 
##          319          320          321          322          323          324 
##  0.839090143  0.739579581  0.686325113  0.675773388  0.831578480  0.341718760 
##          325          326          327          328          329          330 
##  0.290288367  0.168494837 -0.144210596  0.144610634  0.048771048  0.254732871 
##          331          332          333          334          335          336 
##  0.042042209 -0.225549990 -0.236827529 -0.176274543 -0.038612283  0.312138062 
##          337          338          339          340          341          342 
##  0.238171523  0.136415396  0.276010470  0.540565247  0.882857955  0.296840260 
##          343          344          345          346          347          348 
## -0.137283545  1.023863671  1.419062128  1.570806375  1.494699904  1.448212011 
##          349          350          351          352          353          354 
##  0.970092989  1.044901228  1.096022120  1.360122567  1.710005296  1.493208650 
##          355          356          357          358          359          360 
##  1.339123639  1.201445618  1.143456231  0.512135705  1.140535149  0.884675146 
##          361          362          363          364          365          366 
##  1.011324742  0.652221238  0.549047199  0.263673674 -1.126485235 -0.316701781 
##          367          368          369          370          371          372 
##  0.717132716  0.585472735  0.438581794 -0.771989008 -1.290460799 -0.108444124 
##          373          374          375          376          377          378 
##  0.646418665 -0.028111462  0.033091842 -0.250391953 -0.174030337 -0.676824629 
##          379          380          381          382          383          384 
## -0.411439361  0.179887039  0.042274760 -0.135621401 -0.035189170  0.120473964 
##          385          386          387          388          389          390 
##  0.082071971  0.127908668 -0.062952369 -0.241917541  0.150867958 -0.306590869 
##          391          392          393          394          395          396 
## -0.351989173 -0.248284189 -0.229355365  0.066177156 -0.236726148 -0.360486102 
##          397          398          399          400          401          402 
## -0.117368445 -0.335625206 -0.432859367 -0.091015252 -0.352087825 -0.416276302 
##          403          404          405          406          407          408 
## -0.549777331 -0.545873481 -0.383512457 -0.921388416 -0.504774156 -0.748695127 
##          409          410          411          412          413          414 
## -0.782727437 -0.560935127 -0.008560035 -0.664322358 -0.408533211 -0.069387305 
##          415          416          417          418          419          420 
## -0.273464059 -0.132595114 -0.594053109  0.066738616 -0.340858730 -0.257368374 
##          421          422          423          424          425          426 
## -0.414811767 -0.060867515 -0.585942332 -0.240590621 -0.088740505 -0.410675912 
##          427          428          429          430          431          432 
## -0.374591384  0.066396452 -0.154500894 -0.535570467 -0.453401557 -0.172807709 
##          433          434          435          436          437          438 
## -0.458378403 -0.781191169 -0.629485851 -0.904539340 -0.437470823 -0.822157694 
##          439          440          441          442          443          444 
## -0.606324690 -0.105332875 -0.463414756 -0.232166849 -0.367055143 -0.600122567 
##          445          446          447          448          449          450 
## -0.257259980 -0.453475603 -0.286459956 -0.056093756 -0.301833108 -0.395416052 
##          451          452          453          454          455          456 
##  0.025042902  0.097045855  0.140881259 -0.036051822  0.280518057  0.096068539 
##          457          458          459          460          461          462 
## -0.029904984  0.081309895 -0.180521132 -0.067073493  0.033031739  0.025108241 
##          463          464          465          466          467          468 
##  0.216817157 -0.279499964 -0.066774088 -0.114070837 -0.288768122 -0.131827100 
##          469          470          471          472          473          474 
## -0.147972310 -0.398907266 -0.195345702 -0.192634128 -0.193318660  0.072075552 
##          475          476 
## -0.107907378  0.190390635
meck_mod_dl14_res <- ggAcf(residuals(mod_dl14_meck)) + 
  theme_bw(base_size = 15) + ggtitle("")
##            1            2            3            4            5            6 
##  0.083673414  0.203715871  0.108310564  0.045605367  0.110362522  0.393824828 
##            7            8            9           10           11           12 
##  0.283817235  0.228569589 -0.026232304  0.407629201  0.361447059  0.052547076 
##           13           14           15           16           17           18 
##  0.178001206  0.472462403  0.552468387  0.433889572  0.248862452  0.077375063 
##           19           20           21           22           23           24 
##  0.100255268 -0.155851852  0.383149327  0.144611745 -0.141688388 -0.114885685 
##           25           26           27           28           29           30 
## -0.305414029 -0.073615076  0.035925731 -0.252943021 -0.244670279  0.087310528 
##           31           32           33           34           35           36 
## -0.197463150 -0.414951938 -0.149099555 -0.381173178  0.105180877 -0.507462204 
##           37           38           39           40           41           42 
## -0.927093891 -0.222960778 -0.261467804 -0.278147642 -0.227185504 -0.378278193 
##           43           44           45           46           47           48 
##  0.159035080 -0.273853376 -0.092976634 -1.069741022 -0.948109530 -0.306405589 
##           49           50           51           52           53           54 
## -0.863064403  0.480081475  0.778805931  0.711123070  0.895631949  1.175119854 
##           55           56           57           58           59           60 
##  1.470857538  1.673620842  0.877213084  0.798740972  0.606666470  0.926610972 
##           61           62           63           64           65           66 
##  0.822451538  1.109450802  1.085787869  1.182070764  0.857658253  0.731467011 
##           67           68           69           70           71           72 
##  0.488390679  0.108432878  0.324193401  0.536438611  0.288742072 -0.172080777 
##           73           74           75           76           77           78 
##  0.300727820  0.168724681  0.274726918  0.031480759  0.049598504  0.373607464 
##           79           80           81           82           83           84 
## -0.002866966 -0.108282393 -0.176879302  0.005139536  0.039720451  0.006671769 
##           85           86           87           88           89           90 
##  0.082309573  0.012106944  0.195568556  0.117092948 -0.226234898  0.304477535 
##           91           92           93           94           95           96 
## -0.389882718  0.613438625  0.412712740  0.057138967  0.352902740  0.008560608 
##           97           98           99          100          101          102 
##  0.278130924  0.602149876  0.430990641  0.419864616  0.032364319 -0.316782862 
##          103          104          105          106          107          108 
##  0.283565605 -0.175242847 -0.158905309 -0.037251591 -0.110680855 -0.362177583 
##          109          110          111          112          113          114 
## -0.254869641 -0.628388144 -0.436689816 -0.150006003  0.007474222 -0.479450536 
##          115          116          117          118          119          120 
## -0.311955557 -0.741149972 -0.227838354 -0.128741053  0.581673961  0.229427592 
##          121          122          123          124          125          126 
##  0.065854073 -0.227517218 -0.322209225  0.165986256 -0.351316573 -0.412853967 
##          127          128          129          130          131          132 
##  0.703709198  0.141136814 -0.637472279  0.209881186  0.749356210  0.590173817 
##          133          134          135          136          137          138 
##  0.642901903  0.368915261  0.176590063  1.292981938  0.296486825 -0.152654063 
##          139          140          141          142          143          144 
## -0.047852413  0.396782323  0.042512824 -0.140314907 -0.315514338 -0.152180505 
##          145          146          147          148          149          150 
##  0.139575873 -0.356172563  0.071344091  0.330071292  0.465744800  0.647180537 
##          151          152          153          154          155          156 
## -0.240704446 -0.433002635 -0.418524658  0.178443669 -0.202221510 -0.988534265 
##          157          158          159          160          161          162 
## -1.104346842 -0.378046737 -0.599136381 -1.160978629 -0.504870864 -0.218204950 
##          163          164          165          166          167          168 
## -0.997848618 -0.847337339 -0.970523182 -0.058223360 -0.066176988 -0.276425474 
##          169          170          171          172          173          174 
## -0.394131347 -0.348757251 -0.486810964 -0.184599029  0.181121759  0.341363720 
##          175          176          177          178          179          180 
##  0.597635822  0.554918616 -0.046330493  0.344939888  0.135983889  0.157789623 
##          181          182          183          184          185          186 
##  0.172392220  0.035975707  0.697702454  0.251142866  0.314330712  0.462649289 
##          187          188          189          190          191          192 
##  0.554286428  0.404283953  0.432232194  0.454116679  0.405956061  0.254142897 
##          193          194          195          196          197          198 
##  0.506486369  0.639235343  0.173128478  0.317428907  0.357183148  0.097078915 
##          199          200          201          202          203          204 
## -0.145247699 -0.115962461  0.152087391 -0.005357401 -0.211596058 -0.050151630 
##          205          206          207          208          209          210 
##  0.056863765  0.037783605  0.240534616  0.288189768  0.043790436 -0.240101494 
##          211          212          213          214          215          216 
## -0.157707365 -0.056896908  0.125865046  0.109479264  0.211576331  0.394874826 
##          217          218          219          220          221          222 
##  0.324116687  0.368418890  0.408429453  0.104188411  0.054861850  0.144561498 
##          223          224          225          226          227          228 
##  0.321135690  0.158932473  0.006441350  0.161730015 -0.014472257 -0.247536562 
##          229          230          231          232          233          234 
## -0.058038082 -0.174460627 -0.280441817 -0.218221623 -1.236400020 -0.303947449 
##          235          236          237          238          239          240 
## -0.212385134 -0.267329530 -0.187324705 -0.174693433 -0.379732549 -0.198600813 
##          241          242          243          244          245          246 
## -0.355265131 -0.588452008 -0.407572550 -0.358375338 -0.520762780 -0.680479784 
##          247          248          249          250          251          252 
## -0.425806851 -0.490049514 -0.671583984 -0.429451793 -0.435406359 -0.394451973 
##          253          254          255          256          257          258 
## -0.648419830 -0.339791705 -0.554080852 -0.044116526 -0.192096899 -0.179573145 
##          259          260          261          262          263          264 
## -0.367558237 -0.675288139 -0.331304639 -0.313166651 -0.728111349 -0.611805831 
##          265          266          267          268          269          270 
## -0.168956313 -0.458199417 -0.367018295 -0.211525045 -0.361337641 -0.143816302 
##          271          272          273          274          275          276 
## -0.244640382 -0.077397245 -0.029399448 -0.277802980 -0.267612959 -0.450998155 
##          277          278          279          280          281          282 
## -0.408262589 -1.499252712 -1.129477922 -0.112450865  0.039643067 -0.324413390 
##          283          284          285          286          287          288 
## -0.290525830 -0.041874533 -0.459955189  0.027960488  0.297860449 -0.112198979 
##          289          290          291          292          293          294 
## -0.267310416 -0.273467438 -0.776781816 -0.342723799 -0.185990398 -0.008966314 
##          295          296          297          298          299          300 
## -0.075112821 -0.005481946 -0.272122270 -0.122574727 -0.174129075 -0.065889338 
##          301          302          303          304          305          306 
##  0.047714726  0.298739868 -0.025017908  0.289811610  0.217793082  0.239615121 
##          307          308          309          310          311          312 
##  0.239571458  0.414151066  0.257634125  0.204833149  0.506767865  0.406909890 
##          313          314          315          316          317          318 
## -0.057258669  0.482213312  1.231821470  0.721636017  0.728648391  0.500368238 
##          319          320          321          322          323          324 
##  0.731164422  0.642727538  0.907181007  0.502277235  0.234604449  0.227290927 
##          325          326          327          328          329          330 
##  0.041763187  0.034718302 -0.090580190  0.144835121  0.191011942  0.117674808 
##          331          332          333          334          335          336 
##  0.296869566  0.125979818  0.381965504  0.525056634  0.443657128  0.454052683 
##          337          338          339          340          341          342 
##  0.586118672  0.628964184  0.923735461  1.072776733  1.185790856  0.998244704 
##          343          344          345          346          347          348 
##  0.253340933  1.331491217  1.652599841  1.737726257  1.830308586  1.981473344 
##          349          350          351          352          353          354 
##  1.532152911  1.272594809  1.682241025  1.404727965  1.354108842  1.432703322 
##          355          356          357          358          359          360 
##  0.857938160  0.905015563  0.588338824  0.397328873  0.853828081  0.610746309 
##          361          362          363          364          365          366 
##  0.621299482  0.476321744  0.409339184  0.197701568 -1.422533507 -0.616534030 
##          367          368          369          370          371          372 
##  0.223464548  0.300899504  0.100463617 -0.048853561 -0.501220205 -0.265002792 
##          373          374          375          376          377          378 
##  0.038305076 -0.124335803 -0.181012347 -0.264522309 -0.295319384 -0.454387699 
##          379          380          381          382          383          384 
## -0.375987394  0.119548942 -0.173341685  0.010694421 -0.135396856  0.088129302 
##          385          386          387          388          389          390 
## -0.122878115 -0.224254237  0.491590967  0.390222335  0.413354790  0.507118521 
##          391          392          393          394          395          396 
##  0.390756165  0.224829384 -0.395754231  0.394496872  0.131855036 -0.010801313 
##          397          398          399          400          401          402 
##  0.093833413  0.137200493  0.081621831  0.060338738  0.480498679  0.507988234 
##          403          404          405          406          407          408 
##  0.272410677  0.368974209  0.515291904 -0.319724897 -1.206895697 -0.137937622 
##          409          410          411          412          413          414 
## -0.489106405 -0.636590369 -0.282526529 -0.618754014 -1.073271939 -0.120288440 
##          415          416          417          418          419          420 
## -0.796112808 -1.013106913 -0.995653753 -1.527820192 -0.975681290 -0.760436347 
##          421          422          423          424          425          426 
## -1.135932461 -1.105016648 -0.818919288  0.181468417 -0.490653826 -0.725625459 
##          427          428          429          430          431          432 
## -0.903965703 -0.619550178 -0.408564854 -0.701227412 -0.496222007 -0.290672830 
##          433          434          435          436          437          438 
## -0.547162448 -0.890290220 -0.228505158 -1.064244357  0.213677325  0.033515382 
##          439          440          441          442          443          444 
## -0.217047684 -0.711160301 -0.465666053 -0.649149701 -0.309713911 -0.370760781 
##          445          446          447          448          449          450 
## -0.226032639 -0.077406740 -0.409509028 -0.842848432 -0.124986960  0.201279365 
##          451          452          453          454          455          456 
##  0.119362479  0.250029094 -0.013946757 -0.695203647 -0.479920168 -1.362543319 
##          457          458          459          460          461          462 
## -0.437744273 -0.441604822 -0.580462180 -0.474034803 -0.742117255 -0.958541231 
##          463          464          465          466          467          468 
## -0.784336697 -0.334692619 -0.592994945 -0.270884088 -0.603632500 -0.728979911 
##          469          470          471          472          473          474 
## -0.588428646 -0.612960962 -0.473513648 -0.557718658 -0.430461009 -0.501864842 
##          475          476 
## -0.389870632 -0.205309993
hanover_mod_dl14_res <- ggAcf(residuals(mod_dl14_hanover)) + 
  theme_bw(base_size = 15) + ggtitle("")
##            1            2            3            4            5            6 
##  0.747671788  0.307641508  0.378269471  0.473802219  0.619452137  0.711640715 
##            7            8            9           10           11           12 
##  0.784764229  0.851181386  0.587009244  0.494110837  0.452150818  0.244520123 
##           13           14           15           16           17           18 
##  1.122428761  0.583604437  0.912896054  0.215617235  0.691798693  0.600442412 
##           19           20           21           22           23           24 
##  0.524673916  0.873202989  1.163039524  1.087161000  0.647565642  0.586902041 
##           25           26           27           28           29           30 
##  0.787283320  0.752887310  0.961409396  0.688481219  0.380161119  0.573329284 
##           31           32           33           34           35           36 
##  0.443081736  0.358261805  0.027539317  0.293750417 -0.150229915  0.271321532 
##           37           38           39           40           41           42 
##  0.122643563 -0.191472098  0.266413213 -0.044373382  0.366898659  0.576119311 
##           43           44           45           46           47           48 
##  0.673770853  0.442499814  0.353702994  0.518073738  0.721501191  1.016752586 
##           49           50           51           52           53           54 
##  0.968241689  0.797826891  0.564507013  1.403061846  1.371784098  1.189120163 
##           55           56           57           58           59           60 
##  0.831064425  1.273177601  0.966582768  0.844717559  1.023075623  0.807142418 
##           61           62           63           64           65           66 
##  1.018402414  1.150247042  1.277706262  1.143776243  0.986930140  1.396903771 
##           67           68           69           70           71           72 
##  1.453890134  1.287967088  1.252050855  1.127398886  1.351576932  1.109953607 
##           73           74           75           76           77           78 
##  1.439402548  0.826177275  0.798481506  0.559887259  0.878814915  0.460054856 
##           79           80           81           82           83           84 
##  0.618189988  0.354178901  0.800934551  0.644800810  0.272745138  0.556436014 
##           85           86           87           88           89           90 
##  0.698699805  0.765819983  0.676171855  0.678981448  0.541217929  0.878639863 
##           91           92           93           94           95           96 
##  0.480107399 -0.567074083  0.835251043  0.443114773  0.798032497  0.475458293 
##           97           98           99          100          101          102 
##  0.929089500  1.151674303 -0.365146410  1.092744210  0.958868595  0.975938609 
##          103          104          105          106          107          108 
##  0.727230555  0.924896703  0.191074765  0.630066217  0.830326618  0.300237101 
##          109          110          111          112          113          114 
##  0.431653877 -0.470709044  0.587993289  0.713496419  0.170954979  0.864112086 
##          115          116          117          118          119          120 
##  0.419857150  0.941106815  0.304678257  0.619942958  0.000665306  0.049049003 
##          121          122          123          124          125          126 
##  0.532278958 -0.483855497 -0.438958916  0.642657182  0.719306023  0.083623883 
##          127          128          129          130          131          132 
##  0.142308364 -0.547859467  0.679362968 -0.466719588  0.386998573 -0.101381340 
##          133          134          135          136          137          138 
##  0.172972328  0.335315478 -0.049409208 -0.011413372  0.034182159 -0.186190837 
##          139          140          141          142          143          144 
## -0.077516488  0.206178558  0.216607175 -0.276560341 -0.139998257 -0.212428313 
##          145          146          147          148          149          150 
##  0.010084704  0.058939187  0.568749793  0.651043913 -0.375174713 -0.182188461 
##          151          152          153          154          155          156 
## -0.276029586 -0.002888015  0.087513971  0.351314577  0.403895340 -0.027424995 
##          157          158          159          160          161          162 
## -0.004155429  0.089282066  0.116703756 -0.267717173  0.613447150  0.103826154 
##          163          164          165          166          167          168 
## -0.584795668 -0.797745125 -0.923444321 -0.080705315 -0.362428882 -0.193055134 
##          169          170          171          172          173          174 
## -0.022549817 -0.528540196  0.017414671 -0.263990755 -0.635740946 -1.006383371 
##          175          176          177          178          179          180 
## -0.719630746  0.041794680 -0.919307011 -1.240639400  0.153899524 -0.275186941 
##          181          182          183          184          185          186 
##  0.255706766 -0.622651534 -1.010647658 -1.643449840 -0.706496720 -0.554271322 
##          187          188          189          190          191          192 
##  0.085125471 -0.153401590  0.196242232  0.466242979  0.101916302  0.132891233 
##          193          194          195          196          197          198 
## -0.058730517  0.226849953  0.092034675  0.046388800  0.320837843  0.197634154 
##          199          200          201          202          203          204 
##  0.041543478 -0.084591895 -0.031534623  0.376878462  0.083499436  0.131331710 
##          205          206          207          208          209          210 
##  0.125611940  0.160134357 -0.025976111 -0.147004743  0.102707357 -0.132043750 
##          211          212          213          214          215          216 
##  0.142449049  0.222082014 -0.095722886 -0.086447134  0.193431092  0.047369108 
##          217          218          219          220          221          222 
## -0.236510673 -0.011232245  0.630867557  0.965698775  0.842951821  0.980329919 
##          223          224          225          226          227          228 
##  0.405603758 -0.081259496  0.020476796  0.198995617  0.539530739  0.695241902 
##          229          230          231          232          233          234 
##  0.505353089  0.103543071  0.084199783 -0.060773234 -0.687262335  0.223897168 
##          235          236          237          238          239          240 
##  0.158265282 -0.097488959 -0.063130119 -0.320686525 -0.111602996  0.223043923 
##          241          242          243          244          245          246 
##  0.069286926 -0.075827417 -0.195341651  0.007382603 -0.155997697 -0.340909232 
##          247          248          249          250          251          252 
## -0.313957509 -0.278744127 -0.331574026 -0.279316587 -0.669261650 -0.511348060 
##          253          254          255          256          257          258 
## -0.543592370 -0.567951602 -0.649225670 -0.795434355 -1.068118107 -0.536898969 
##          259          260          261          262          263          264 
## -0.781653867 -1.378215169 -0.642796987 -0.695479334 -1.133816241 -0.705216939 
##          265          266          267          268          269          270 
## -0.833183745 -0.481082101 -0.949364510 -0.666934935 -0.729533596 -0.670244008 
##          271          272          273          274          275          276 
## -0.459775767 -0.312795933 -0.683795518 -0.958265199 -0.710554591 -0.461965690 
##          277          278          279          280          281          282 
## -1.927703353 -1.860354405 -1.788623255 -0.580177525 -1.364166882 -1.264233742 
##          283          284          285          286          287          288 
## -1.405559426 -0.994684914 -1.846049333 -1.169577240 -1.522802689 -1.370957154 
##          289          290          291          292          293          294 
## -1.488322029 -0.898873445 -1.182528558 -1.869183395 -1.303331789 -0.838020471 
##          295          296          297          298          299          300 
## -0.783640584 -0.184541374 -0.804762372 -0.845862677 -0.639393492 -0.073034083 
##          301          302          303          304          305          306 
##  0.505106704 -0.414760325 -0.152886163  0.011629339  0.175510337  0.008692918 
##          307          308          309          310          311          312 
## -0.466941362 -0.303730275  0.212979426 -0.354538102 -0.630846915 -0.610448748 
##          313          314          315          316          317          318 
## -1.751952004 -1.272773337 -0.374306882  0.290963727  0.046446696 -0.214981661 
##          319          320          321          322          323          324 
## -0.545341288 -0.332527264 -0.469964243 -0.455682589 -0.018678859 -0.219464918 
##          325          326          327          328          329          330 
## -0.255516321  0.299562431  0.462774125 -0.712331746  0.204310949  0.378700005 
##          331          332          333          334          335          336 
##  0.236595198  0.131521713  0.023084524  0.309143039  0.393019139  0.514684828 
##          337          338          339          340          341          342 
##  0.709765643  0.815688120  0.422194522  0.927938263  1.026089235  0.119582660 
##          343          344          345          346          347          348 
## -0.400948873  0.618876707  1.283429018  1.428341833  1.416771338  1.546686323 
##          349          350          351          352          353          354 
##  0.844014152  0.520292465  0.964667125  1.468735348  1.440001829  1.088767430 
##          355          356          357          358          359          360 
##  1.381710323  1.285408988  0.927875505  0.875859442  1.146208669  1.053124920 
##          361          362          363          364          365          366 
##  0.992434881  1.103931611  1.235399280  0.792817833  0.536064069  1.022193226 
##          367          368          369          370          371          372 
##  1.369988900  1.386744406  1.145528674  0.111164949 -0.122587151  0.403142012 
##          373          374          375          376          377          378 
##  0.917741437  0.574358546  0.414126633  0.390922055  0.409998132 -0.047921889 
##          379          380          381          382          383          384 
##  0.051743407  0.346350681  0.219834318  0.183649959  0.003918620 -0.087689089 
##          385          386          387          388          389          390 
## -0.443879508 -0.368255592 -0.233817769 -0.134835441 -0.394255333 -0.319883083 
##          391          392          393          394          395          396 
## -0.218698343 -0.910699733 -0.465972929 -0.585333047 -0.694596814 -0.586108231 
##          397          398          399          400          401          402 
## -0.738585332 -0.679538577 -1.118610524 -0.420766359 -0.638185474 -0.714555809 
##          403          404          405          406          407          408 
## -0.585440652 -0.563505308 -0.878102761 -1.280603695 -1.191681077 -0.934102963 
##          409          410          411          412          413          414 
## -1.596363594 -1.557809531 -0.722601433 -1.255975639 -0.770097847 -0.727531531 
##          415          416          417          418          419          420 
## -0.518165467 -1.293445438 -0.778215674 -0.565879295 -0.856861437 -0.400860881 
##          421          422          423          424          425          426 
## -0.440988575 -0.959018978 -0.689239873 -0.689466316 -0.532029981 -0.278728684 
##          427          428          429          430          431          432 
##  0.266328315  0.783612944 -0.362063857 -0.402825267 -0.403869896 -0.366283882 
##          433          434          435          436          437          438 
## -0.063202851  0.165020700  0.151977809 -0.472627095 -0.542580764 -0.820343633 
##          439          440          441          442          443          444 
## -0.583075855 -0.521115134 -0.403670372 -0.526089683 -0.566016826 -1.104702309 
##          445          446          447          448          449          450 
## -0.806441249 -0.082094347 -0.979519688 -0.878787309 -0.900340594 -0.696046618 
##          451          452          453          454          455          456 
## -0.459637216  0.026626468 -0.546385035 -0.882709570 -1.394742797 -1.432171235 
##          457          458          459          460          461          462 
## -0.853846173 -0.715528872 -1.211671395 -1.384127172 -1.836427085 -0.407158641 
##          463          464          465          466          467          468 
## -1.607260055 -0.880169779 -0.844179745 -0.666642593 -0.660131954 -0.278294858 
##          469          470          471          472          473          474 
## -1.266666170 -1.041434986 -0.633569678 -0.426953761 -0.315162411 -0.559516785 
##          475          476 
## -0.196315618 -0.496435946
png(filename = "DL_res.png", units = "cm", res = 700, 
    width = 20, height = 15)
grid.arrange(wake_mod_dl14_res,
             meck_mod_dl14_res,
             hanover_mod_dl14_res)
dev.off()
## quartz_off_screen 
##                 2
#DL forecasting plots#

full_cases_wastewater_weather_data_test <-
  cbind(full_cases_wastewater_weather_data_test,f_dl14$forecasts[,2],
        f_dl14$forecasts[,1],f_dl14$forecasts[,3])

full_cases_wastewater_weather_data_meck_test <-
  cbind(full_cases_wastewater_weather_data_meck_test,f_dl14_meck$forecasts[,2],
        f_dl14_meck$forecasts[,1],f_dl14_meck$forecasts[,3])

full_cases_wastewater_weather_data_hanover_test <-
  cbind(full_cases_wastewater_weather_data_hanover_test,f_dl14_hanover$forecasts[,2],
        f_dl14_hanover$forecasts[,1],f_dl14_hanover$forecasts[,3])


wake_dl_plot <- full_cases_wastewater_weather_data_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_test, aes(ymin = f_dl14$forecasts[,1], ymax = f_dl14$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_test,aes(Date,f_dl14$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

meck_dl_plot <- full_cases_wastewater_weather_data_meck_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_meck_test, aes(ymin = f_dl14_meck$forecasts[,1], ymax = f_dl14_meck$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_meck_test,aes(Date,f_dl14_meck$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")

hanover_dl_plot <- full_cases_wastewater_weather_data_hanover_train %>% 
  ggplot(aes(Date,log_mean_new_cases)) + 
  geom_line() + 
  geom_ribbon(data = full_cases_wastewater_weather_data_hanover_test, aes(ymin = f_dl14_hanover$forecasts[,1], ymax = f_dl14_hanover$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,log_mean_new_cases,color="Actual")) +
  geom_line(data = full_cases_wastewater_weather_data_hanover_test,aes(Date,f_dl14_hanover$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "bottom") + ylab("")

png(filename = "dl_plots.png",res = 700, units = "cm",
    width = 20,height = 15)
grid.arrange(wake_dl_plot,
             meck_dl_plot,
             hanover_dl_plot,
             ncol=1,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2

Sensitivity analysis

#Cases

cases <- read.csv("cases_by_county.csv")
glimpse(cases)
## Rows: 158,340
## Columns: 6
## $ X                                       <int> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …
## $ date                                    <chr> "2021-11-20", "2021-11-23", "2…
## $ rolling_average_cases_per_100k_centered <dbl> 58.628121, 53.867485, 46.16043…
## $ state                                   <chr> "AZ", "AZ", "AZ", "AZ", "AZ", …
## $ name                                    <chr> "Pima County, AZ", "Pima Count…
## $ fipscode                                <int> 4019, 4019, 4019, 4019, 4019, …
cases <- cases[order(as.Date(cases$date)),]
cases <- cases[cases$date >= "2021-01-04" &
                 cases$date <= "2022-05-22",]
cases_nc <- cases %>%  dplyr::filter(state == "NC")
glimpse(cases_nc)
## Rows: 1,512
## Columns: 6
## $ X                                       <int> 56308, 103759, 116281, 16478, …
## $ date                                    <chr> "2021-01-04", "2021-01-04", "2…
## $ rolling_average_cases_per_100k_centered <dbl> 63.71797, 101.44618, 55.99169,…
## $ state                                   <chr> "NC", "NC", "NC", "NC", "NC", …
## $ name                                    <chr> "Duplin County, NC", "Stanly C…
## $ fipscode                                <int> 37061, 37167, 37051, 37167, 37…
cases_nc <- cases_nc %>% group_by(date) %>% summarise(mean_cases=
                                            mean(rolling_average_cases_per_100k_centered))

#Wastewater

wastewater <- read.csv("wastewater_by_county.csv")
glimpse(wastewater)
## Rows: 6,562
## Columns: 7
## $ X                                       <int> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …
## $ sampling_week                           <chr> "2020-01-01", "2020-01-15", "2…
## $ effective_concentration_rolling_average <dbl> 134.841964, 0.000000, 0.000000…
## $ region                                  <chr> "Midwest", "Northeast", "North…
## $ state                                   <chr> "IL", "MA", "MA", "MA", "MA", …
## $ name                                    <chr> "Peoria County, IL", "Suffolk …
## $ fipscode                                <int> 17143, 25025, 25025, 25025, 25…
wastewater <- wastewater[order(as.Date(wastewater$sampling_week)),]
wastewater <- wastewater[wastewater$sampling_week >= "2021-01-04" &
                           wastewater$sampling_week <= "2022-05-22",]
wastewater_nc <- wastewater %>%  dplyr::filter(state == "NC")
glimpse(wastewater_nc)
## Rows: 38
## Columns: 7
## $ X                                       <int> 1662, 1735, 1739, 1825, 1830, …
## $ sampling_week                           <chr> "2021-06-09", "2021-06-16", "2…
## $ effective_concentration_rolling_average <dbl> 162.00024, 305.01615, 81.08650…
## $ region                                  <chr> "South", "South", "South", "So…
## $ state                                   <chr> "NC", "NC", "NC", "NC", "NC", …
## $ name                                    <chr> "Cumberland County, NC", "Cumb…
## $ fipscode                                <int> 37051, 37051, 37061, 37051, 37…
wastewater_nc <- wastewater_nc %>% group_by(sampling_week) %>% 
  summarise(mean_wastewater = mean(effective_concentration_rolling_average))
wastewater_nc$sampling_week <- as.Date(wastewater_nc$sampling_week)
wastewater_nc <- pad(wastewater_nc,start_val=as.Date("2021-01-04"),
                     end_val=as.Date("2022-05-22"))
wastewater_nc <- wastewater_nc %>% 
  mutate(mean_wastewater = na_locf(mean_wastewater,))
colnames(wastewater_nc)[1] <-"date"

#merge dataset

nc_data <- cbind(cases_nc,wastewater_nc[,-1])
glimpse(nc_data)
## Rows: 504
## Columns: 3
## $ date            <chr> "2021-01-04", "2021-01-05", "2021-01-06", "2021-01-07"…
## $ mean_cases      <dbl> 73.71861, 89.17613, 79.05748, 83.65169, 82.17460, 83.8…
## $ mean_wastewater <dbl> 162.0002, 162.0002, 162.0002, 162.0002, 162.0002, 162.…
nc_data$date <- as.Date(nc_data$date)

#Line plots

png(filename = "line_plot_sensitivity_analysis.png",units = "cm",res = 700,
    width = 20, height = 14)
case_plot <- nc_data %>% ggplot(aes(date,mean_cases)) + geom_line() +
  ylab("COVID-19 case counts per 100k") + theme_bw()
ww_plot <- nc_data %>% ggplot(aes(date,mean_wastewater)) + geom_line() +
  ylab("Viral gene copies per mL") + theme_bw()
grid.arrange(case_plot,
             ww_plot)
dev.off()
## quartz_off_screen 
##                 2
#Correlations

png(filename = "nc_cor.png",units = "cm",res = 700,
    width = 20, height = 8)
nc_data %>% ggplot(aes(mean_wastewater,mean_cases)) + geom_point() +
  theme_bw() + xlab("Viral gene copies per mL") + ylab("New Cases")
dev.off()
## quartz_off_screen 
##                 2
cor.test(nc_data$mean_cases,nc_data$mean_wastewater)
## 
##  Pearson's product-moment correlation
## 
## data:  nc_data$mean_cases and nc_data$mean_wastewater
## t = 14.884, df = 502, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4896649 0.6111462
## sample estimates:
##       cor 
## 0.5533411
#arima

cases <- xts(nc_data$mean_cases, order.by = nc_data$date)
attr(cases, 'frequency') <- 7 
periodicity(cases)  
## Daily periodicity from 2021-01-04 to 2022-05-22
cases  <- as.ts(cases)
#plot(decompose(log(cases)))

cases_deseasonalise <- seasadj(decompose(log(cases)))

cases_deseasonalise_train <- cases_deseasonalise[-c(491:504)]
cases_deseasonalise_test <- cases_deseasonalise[c(491:504)]

lowest_rmse<-Inf
lowest_mae<-Inf
best_mod<-NULL
best_mod_mae<-NULL

for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(cases_deseasonalise_train, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    rmse_mod_1 <- rmse(cases_deseasonalise_test,forecast_fit$mean)
    if (rmse_mod_1 < lowest_rmse){
      lowest_rmse <- rmse_mod_1 
      best_mod <- arima_mod_1
    }
  }
}

lowest_rmse
## [1] 0.3455963
best_mod #arima(3,1,1)
## Series: cases_deseasonalise_train 
## ARIMA(3,1,1) 
## 
## Coefficients:
##          ar1     ar2     ar3      ma1
##       0.6336  0.1419  0.1348  -0.8879
## s.e.  0.0746  0.0541  0.0456   0.0620
## 
## sigma^2 = 0.03666:  log likelihood = 116.4
## AIC=-222.8   AICc=-222.67   BIC=-201.84
for (p in seq(1:4)){
  for (q in seq(1:4)){
    arima_mod_1 <- Arima(cases_deseasonalise_train, order = c(p,1,q))
    forecast_fit <- forecast::forecast(arima_mod_1,h=14)
    mae_mod <- mae(cases_deseasonalise_test,forecast_fit$mean)
    if (mae_mod < lowest_mae){
      lowest_mae <- mae_mod
      best_mod_mae <- arima_mod_1
    }
  }
} 

best_mod_mae
## Series: cases_deseasonalise_train 
## ARIMA(3,1,1) 
## 
## Coefficients:
##          ar1     ar2     ar3      ma1
##       0.6336  0.1419  0.1348  -0.8879
## s.e.  0.0746  0.0541  0.0456   0.0620
## 
## sigma^2 = 0.03666:  log likelihood = 116.4
## AIC=-222.8   AICc=-222.67   BIC=-201.84
lowest_mae #arima(3,1,1)
## [1] 0.2671189
best_arima_mod <- Arima(cases_deseasonalise_train, order = c(3,1,1))
best_arima_mod_forecast <- forecast::forecast(best_arima_mod,h=14)
rmse(cases_deseasonalise_test,best_arima_mod_forecast$mean) 
## [1] 0.3455963
mae(cases_deseasonalise_test,best_arima_mod_forecast$mean) 
## [1] 0.2671189
checkresiduals(best_arima_mod)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(3,1,1)
## Q* = 32.905, df = 6, p-value = 1.093e-05
## 
## Model df: 4.   Total lags used: 10
png(filename = "sensitivity_nc_arima.png",res = 700, units = "cm",
    width = 20, height = 10)
arima_plot <- autoplot(best_arima_mod_forecast) + 
  autolayer(best_arima_mod_forecast, series = "Forecasted") +
  autolayer(ts(cases_deseasonalise_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time")+
  ggtitle(NULL) + theme(legend.position = "none") 
dev.off()
## quartz_off_screen 
##                 2
exp(cases_deseasonalise_test[1]) 
## [1] 11.68326
exp(best_arima_mod_forecast$mean[1]) 
## [1] 12.25043
exp(best_arima_mod_forecast$lower[1,]) 
##      80%      95% 
## 9.584871 8.417325
exp(best_arima_mod_forecast$upper[1,]) 
##      80%      95% 
## 15.65728 17.82906
exp(best_arima_mod_forecast$mean[1])-exp(cases_deseasonalise_test[1])
## [1] 0.5671695
exp(cases_deseasonalise_test[7])
## [1] 18.14823
exp(best_arima_mod_forecast$mean[7])
## [1] 12.36624
exp(best_arima_mod_forecast$lower[7,])
##      80%      95% 
## 7.170108 5.373013
exp(best_arima_mod_forecast$upper[7,])
##      80%      95% 
## 21.32798 28.46147
exp(best_arima_mod_forecast$mean[7])-exp(cases_deseasonalise_test[7])
## [1] -5.781993
exp(cases_deseasonalise_test[14])
## [1] 25.32937
exp(best_arima_mod_forecast$mean[14])
## [1] 12.45729
exp(best_arima_mod_forecast$lower[14,])
##      80%      95% 
## 5.440607 3.509080
exp(best_arima_mod_forecast$upper[14,])
##      80%      95% 
## 28.52332 44.22360
exp(best_arima_mod_forecast$mean[14])-exp(cases_deseasonalise_test[14])
## [1] -12.87208
#SARIMA

cases_train <- log(cases)[-c(491:504)]
cases_test <- log(cases)[c(491:504)]

sarima_rmse <- Inf
sarima_best_mod <-NULL

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(cases_train, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            rsme_mod <- rmse(cases_test,forecast_fit$mean)
            if(rsme_mod<sarima_rmse){
              sarima_rmse<- rsme_mod 
              sarima_best_mod<-sarima_mod_1 
            }
          }
        }
      }
    }
  }
}

sarima_rmse
## [1] 0.04706877
sarima_best_mod #arima(0,3,3)(1,3,2)[7]
## Series: cases_train 
## ARIMA(0,3,3)(1,3,2)[7] 
## 
## Coefficients:
##           ma1     ma2      ma3     sar1     sma1    sma2
##       -2.2331  1.5617  -0.3285  -0.7234  -1.7973  0.8678
## s.e.      NaN     NaN      NaN      NaN   0.0180     NaN
## 
## sigma^2 = 0.08438:  log likelihood = -85.66
sarima_mae <- Inf
sarima_best_mod_mae <-NULL

for (p in seq(0,3)){
  for (d in seq(0,3)){
    for (q in seq(0,3)){
      for (P in seq(0,3)){
        for (D in seq(0,3)){
          for (Q in seq(0,3)){
            
            sarima_mod_1 <- Arima(cases_train, order = c(p,d,q),
                                  seasonal = list(order=c(P,D,Q),period=7),
                                  method="CSS")
            forecast_fit <- forecast::forecast(sarima_mod_1,14)
            mae_mod <- mae(cases_test,forecast_fit$mean)
            if(mae_mod<sarima_mae){
              sarima_mae<- mae_mod 
              sarima_best_mod_mae<-sarima_mod_1 
            }
          }
        }
      }
    }
  }
}

sarima_mae 
## [1] 0.03468064
sarima_best_mod_mae 
## Series: cases_train 
## ARIMA(0,3,2)(3,2,2)[7] 
## 
## Coefficients:
##           ma1     ma2     sar1     sar2     sar3     sma1    sma2
##       -1.9398  0.9407  -0.6269  -0.3011  -0.0470  -1.3960  0.4249
## s.e.   0.0023  0.0024   0.1282   0.1152   0.0174   0.1607  0.1631
## 
## sigma^2 = 0.05142:  log likelihood = 23.5
sarima_best_mod_1 <- Arima(cases_train, order = c(0,3,3),
                           seasonal = list(order=c(1,3,2),period=7),
                           method="CSS")
sarima_best_mod_1_forecast <- forecast::forecast(sarima_best_mod_1,14)
rmse(cases_test,sarima_best_mod_1_forecast$mean)
## [1] 0.04706877
mae(cases_test,sarima_best_mod_1_forecast$mean)
## [1] 0.03691424
checkresiduals(sarima_best_mod_1)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,3,3)(1,3,2)[7]
## Q* = 8.9851, df = 4, p-value = 0.06147
## 
## Model df: 6.   Total lags used: 10
sarima_best_mod_3 <- Arima(cases_train, order = c(0,3,2),
                           seasonal = list(order=c(3,2,2),period=7),
                           method="CSS")
coeftest(sarima_best_mod_3)
## 
## z test of coefficients:
## 
##        Estimate Std. Error   z value  Pr(>|z|)    
## ma1  -1.9398426  0.0022785 -851.3765 < 2.2e-16 ***
## ma2   0.9407284  0.0024320  386.8186 < 2.2e-16 ***
## sar1 -0.6268993  0.1281771   -4.8909 1.004e-06 ***
## sar2 -0.3010714  0.1152230   -2.6129  0.008977 ** 
## sar3 -0.0469942  0.0173562   -2.7076  0.006776 ** 
## sma1 -1.3959982  0.1606967   -8.6872 < 2.2e-16 ***
## sma2  0.4249349  0.1630928    2.6055  0.009175 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
sarima_best_mod_3_forecast <- forecast::forecast(sarima_best_mod_3,14)
rmse(cases_test,sarima_best_mod_3_forecast$mean) 
## [1] 0.04984621
mae(cases_test,sarima_best_mod_3_forecast$mean) 
## [1] 0.03468064
checkresiduals(sarima_best_mod_3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,3,2)(3,2,2)[7]
## Q* = 32.82, df = 3, p-value = 3.514e-07
## 
## Model df: 7.   Total lags used: 10
png(filename = "sensitivity_nc_sarima_simple.png",res = 700, units = "cm",
    width = 20, height = 10)
sarima_plot <- autoplot(sarima_best_mod_3_forecast) + 
  autolayer(sarima_best_mod_3_forecast, series = "Forecasted") +
  autolayer(ts(cases_deseasonalise_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time")+
  ggtitle(NULL) + theme(legend.position = "none") 
dev.off()
## quartz_off_screen 
##                 2
exp(cases_deseasonalise_test[1]) 
## [1] 11.68326
exp(sarima_best_mod_3_forecast$mean[1]) 
## [1] 11.80592
exp(sarima_best_mod_3_forecast$lower[1,])
##      80%      95% 
## 8.831264 7.573240
exp(sarima_best_mod_3_forecast$upper[1,]) 
##      80%      95% 
## 15.78254 18.40425
exp(sarima_best_mod_3_forecast$mean[1])-exp(cases_test[1])
## [1] -0.08975849
exp(cases_deseasonalise_test[7])
## [1] 18.14823
exp(sarima_best_mod_3_forecast$mean[7])
## [1] 17.80267
exp(sarima_best_mod_3_forecast$lower[7,])
##      80%      95% 
## 7.127076 4.389838
exp(sarima_best_mod_3_forecast$upper[7,])
##      80%      95% 
## 44.46918 72.19747
exp(sarima_best_mod_3_forecast$mean[7])-exp(cases_test[7])
## [1] -0.8412984
exp(cases_deseasonalise_test[14])
## [1] 25.32937
exp(sarima_best_mod_3_forecast$mean[14])
## [1] 23.16659
exp(sarima_best_mod_3_forecast$lower[14,])
##      80%      95% 
## 4.934803 2.176459
exp(sarima_best_mod_3_forecast$upper[14,])
##      80%      95% 
## 108.7563 246.5891
exp(sarima_best_mod_3_forecast$mean[14])-exp(cases_test[14])
## [1] -2.854685
#multivariate modelling#

#ARIMAX, with water

viral_deseasonalise <- seasadj(decompose(ts(log(nc_data$mean_wastewater),
                                            frequency = 7)))

viral_deseasonalise_train <- viral_deseasonalise[-c(491:504)]
viral_deseasonalise_test <- viral_deseasonalise[c(491:504)]

wastewater_mod <- Arima(cases_deseasonalise_train ,order = c(3,1,1),
                        xreg = viral_deseasonalise_train)
coeftest(wastewater_mod) 
## 
## z test of coefficients:
## 
##       Estimate Std. Error  z value  Pr(>|z|)    
## ar1   0.632174   0.072198   8.7561 < 2.2e-16 ***
## ar2   0.142602   0.053842   2.6485  0.008085 ** 
## ar3   0.137426   0.045916   2.9930  0.002763 ** 
## ma1  -0.886679   0.059077 -15.0089 < 2.2e-16 ***
## xreg -0.048612   0.082454  -0.5896  0.555482    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
forecast_mod_1 <- forecast::forecast(wastewater_mod, h=14,
                                     xreg = viral_deseasonalise_test) 
rmse(cases_deseasonalise_test,forecast_mod_1$mean) 
## [1] 0.3456571
mae(cases_deseasonalise_test,forecast_mod_1$mean) 
## [1] 0.2662839
tsdisplay(residuals(wastewater_mod)) 

png(filename = "sensitivity_nc_arimax.png",res = 700, units = "cm",
    width = 20, height = 10)
arimax_plot <- autoplot(forecast_mod_1) + 
  autolayer(forecast_mod_1 , series = "Forecasted") +
  autolayer(ts(cases_deseasonalise_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time") +
  ggtitle(NULL) + theme(legend.position = "none")
dev.off()
## quartz_off_screen 
##                 2
exp(cases_deseasonalise_test[1]) 
## [1] 11.68326
exp(forecast_mod_1$mean[1]) 
## [1] 12.2664
exp(forecast_mod_1$lower[1,]) 
##      80%      95% 
## 9.595798 8.426193
exp(forecast_mod_1$upper[1,]) 
##      80%      95% 
## 15.68025 17.85677
exp(forecast_mod_1$mean[1])-exp(cases_deseasonalise_test[1])
## [1] 0.5831395
exp(cases_deseasonalise_test[7])
## [1] 18.14823
exp(forecast_mod_1$mean[7])
## [1] 12.32183
exp(forecast_mod_1$lower[7,])
##      80%      95% 
## 7.126134 5.332845
exp(forecast_mod_1$upper[7,])
##      80%      95% 
## 21.30574 28.47027
exp(forecast_mod_1$mean[7])-exp(cases_deseasonalise_test[7])
## [1] -5.8264
exp(cases_deseasonalise_test[14])
## [1] 25.32937
exp(forecast_mod_1$mean[14])
## [1] 12.47918
exp(forecast_mod_1$lower[14,])
##      80%      95% 
## 5.404005 3.469813
exp(forecast_mod_1$upper[14,])
##      80%      95% 
## 28.81751 44.88136
exp(forecast_mod_1$mean[14])-exp(cases_deseasonalise_test[14])
## [1] -12.85019
#SARIMAX, with water

viral_train <- log(nc_data$mean_wastewater)[-c(491:504)]
viral_test <- log(nc_data$mean_wastewater)[c(491:504)]

sarimax_wastewater_mod <- Arima(cases_train ,order = c(0,3,2),
                               seasonal = list(order=c(3,2,2),period=7),
                               xreg = viral_train,
                               method = "CSS-ML")
coeftest(sarimax_wastewater_mod) 
## 
## z test of coefficients:
## 
##        Estimate Std. Error   z value Pr(>|z|)    
## ma1  -1.9601395  0.0032763 -598.2782   <2e-16 ***
## ma2   0.9604238  0.0032989  291.1352   <2e-16 ***
## sar1 -0.5922529  0.0014546 -407.1487   <2e-16 ***
## sar2 -0.2887453  0.0035108  -82.2441   <2e-16 ***
## sar3 -0.0703806  0.0042554  -16.5391   <2e-16 ***
## sma1 -1.4521516  0.0031877 -455.5529   <2e-16 ***
## sma2  0.4574305  0.0096941   47.1866   <2e-16 ***
## xreg -0.0323508  0.0799103   -0.4048   0.6856    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
sarimax_wastewater_mod_forecast <- forecast::forecast(sarimax_wastewater_mod, h=14,
                                                     xreg = viral_test) 
rmse(cases_test,sarimax_wastewater_mod_forecast$mean) 
## [1] 0.06608591
mae(cases_test,sarimax_wastewater_mod_forecast$mean) 
## [1] 0.05076526
checkresiduals(sarimax_wastewater_mod) 

## 
##  Ljung-Box test
## 
## data:  Residuals from Regression with ARIMA(0,3,2)(3,2,2)[7] errors
## Q* = 36.206, df = 3, p-value = 6.773e-08
## 
## Model df: 8.   Total lags used: 11
png(filename = "sensitivity_nc_sarimax_complex.png",res = 700, units = "cm",
    width = 20, height = 10)
sarimax_plot <- autoplot(sarimax_wastewater_mod_forecast) + 
  autolayer(sarimax_wastewater_mod_forecast, series = "Forecasted") +
  autolayer(ts(cases_test,start = 491), series = "Observed") +
  theme_bw(base_size = 15) + ylab("") + xlab("Time") +
  ggtitle(NULL) + theme(legend.position = "none") 
dev.off()
## quartz_off_screen 
##                 2
exp(sarimax_wastewater_mod_forecast$mean[1]) 
## [1] 11.76755
exp(sarimax_wastewater_mod_forecast$lower[1,]) 
##      80%      95% 
## 8.913452 7.694546
exp(sarimax_wastewater_mod_forecast$upper[1,]) 
##      80%      95% 
## 15.53552 17.99653
exp(sarimax_wastewater_mod_forecast$mean[1])-exp(cases_test[1])
## [1] -0.1281357
exp(sarimax_wastewater_mod_forecast$mean[7])
## [1] 17.19558
exp(sarimax_wastewater_mod_forecast$lower[7,])
##      80%      95% 
## 7.484325 4.818489
exp(sarimax_wastewater_mod_forecast$upper[7,])
##      80%      95% 
## 39.50765 61.36531
exp(sarimax_wastewater_mod_forecast$mean[7])-exp(cases_test[7])
## [1] -1.448389
exp(sarimax_wastewater_mod_forecast$mean[14])
## [1] 22.66054
exp(sarimax_wastewater_mod_forecast$lower[14,])
##      80%      95% 
## 6.037109 2.997355
exp(sarimax_wastewater_mod_forecast$upper[14,])
##       80%       95% 
##  85.05725 171.31765
exp(sarimax_wastewater_mod_forecast$mean[14])-exp(cases_test[14])
## [1] -3.360739
#Autoregressive Distributed Lag Model

nc_data <- nc_data %>%
  mutate(log_cases = log(mean_cases),
         log_viral = log(mean_wastewater))

nc_data <- nc_data %>%
  mutate(log_cases = seasadj(decompose(ts(log_cases,frequency = 7))),
         log_viral = seasadj(decompose(ts(log_viral,frequency = 7))))

nc_data_train <- nc_data[-c(491:504),]
nc_data_test <- nc_data[c(491:504),]

lowest_rmse_ardl <- Inf
best_mod_ardl <- NULL

for (p in seq(1,14)){
  for (q in seq(1,14)){
    mod <- ardlDlm(log_cases ~ log_viral,
                   data = nc_data_train, p=p,q=q)
    f <- forecast(mod, x= t(nc_data_test[,5]),h=14)
    forecast_acc <- rmse(nc_data_test[,4],
                         f$forecasts)
    if (forecast_acc<lowest_rmse_ardl){
      lowest_rmse_ardl<- forecast_acc
      best_mod_ardl <- mod 
    }
  }
}

lowest_rmse_ardl #0.153
## [1] 0.1527168
summary(best_mod_ardl) #ardl(1,14), similar to wake
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.90992 -0.04728 -0.00450  0.03391  1.82292 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.0510420  0.0547900  -0.932  0.35204    
## log_viral.t   0.0068198  0.0962389   0.071  0.94354    
## log_viral.1  -0.0004429  0.1355423  -0.003  0.99739    
## log_viral.2  -0.0662456  0.1355423  -0.489  0.62526    
## log_viral.3  -0.0267071  0.1355444  -0.197  0.84389    
## log_viral.4   0.3559041  0.1355462   2.626  0.00894 ** 
## log_viral.5  -0.2269657  0.1359995  -1.669  0.09582 .  
## log_viral.6  -0.0071780  0.1363630  -0.053  0.95804    
## log_viral.7  -0.0289224  0.1364897  -0.212  0.83228    
## log_viral.8  -0.0003262  0.1361771  -0.002  0.99809    
## log_viral.9   0.1139565  0.1361771   0.837  0.40313    
## log_viral.10 -0.0872283  0.1361834  -0.641  0.52215    
## log_viral.11  0.0869370  0.1361774   0.638  0.52353    
## log_viral.12 -0.0677954  0.1362716  -0.498  0.61907    
## log_viral.13  0.0133665  0.1363593   0.098  0.92196    
## log_viral.14 -0.0393125  0.0967766  -0.406  0.68477    
## log_cases.1   0.9662545  0.0113959  84.790  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1981 on 459 degrees of freedom
## Multiple R-squared:  0.9661, Adjusted R-squared:  0.9649 
## F-statistic: 818.1 on 16 and 459 DF,  p-value: < 2.2e-16
checkresiduals(best_mod_ardl)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
## -0.0079361204  0.0749551140  0.0104102776  0.0417587039  0.1788979741 
##            20            21            22            23            24 
## -0.0197832873 -0.0636556796  0.0659424356 -0.0058961325  0.0563267039 
##            25            26            27            28            29 
##  0.0722532860  0.0555601394  0.0195105611  0.1191322887  0.0569816684 
##            30            31            32            33            34 
##  0.0432000544  0.0106782259  0.0277613061  0.0608521145  0.0164457056 
##            35            36            37            38            39 
## -0.1385073028  0.0055273619  0.0465037029  0.0195600474 -0.0073049436 
##            40            41            42            43            44 
##  0.0448363228  0.0049317851 -0.0320415286  0.0338699326  0.0305961503 
##            45            46            47            48            49 
##  0.0283465099  0.0926974719  0.0618128861 -0.0184534334  0.0213206569 
##            50            51            52            53            54 
##  0.0326426656 -0.0040375544 -0.0371331073 -0.1782426977  0.2204057225 
##            55            56            57            58            59 
##  0.0104794007 -0.0693380908 -0.0828500659  0.0462878989  0.0246169661 
##            60            61            62            63            64 
##  0.0262632746 -0.0843961025 -0.0379274320 -0.0720000488  0.0817978511 
##            65            66            67            68            69 
## -0.0246907811 -0.1745607217  0.0117215188  0.2851422174  0.0443775771 
##            70            71            72            73            74 
##  0.0028641604 -0.0653787188  0.0414945238  0.1727227919  0.0492103829 
##            75            76            77            78            79 
## -0.1762951948 -0.0672963539 -0.0814167277  0.1228827473  0.0513663051 
##            80            81            82            83            84 
## -0.1715105563 -0.0272278816  0.3267267423  0.0269302279 -0.0049594610 
##            85            86            87            88            89 
## -0.0068720516 -0.1771227030  0.0139272167  0.0124382865  0.0116317455 
##            90            91            92            93            94 
## -0.0586408875 -0.0057825049 -0.0022346363  0.2057315708  0.0174397517 
##            95            96            97            98            99 
##  0.0159508215  0.0837310328  0.0630946687 -0.0019086285  0.0872437669 
##           100           101           102           103           104 
##  0.0692130295  0.0246579464  0.0231690162  0.0438558048 -0.0308327461 
##           105           106           107           108           109 
## -0.0682791666  0.0288945455  0.0165520992  0.0209345474  0.0194456172 
##           110           111           112           113           114 
##  0.0187354925  0.0234234992  0.0130892478  0.0212016080  0.0359539623 
##           115           116           117           118           119 
##  0.0216787007  0.0201897705 -0.0124944339  0.0329172523 -0.0371419702 
##           120           121           122           123           124 
## -0.0537262902  0.0013843828  0.0164835277  0.0149945975  0.0220603837 
##           125           126           127           128           129 
## -0.1017259910  0.0188264098 -0.0081172809 -0.0540654425  0.0104045561 
##           130           131           132           133           134 
##  0.0089156258 -0.0880293066  0.0128322586 -0.1653236580  0.0155740275 
##           135           136           137           138           139 
## -0.0111191180  0.0019533161  0.0004643858 -0.1051715333 -0.0873028623 
##           140           141           142           143           144 
## -0.1879690028 -0.0661025606 -0.0561235139 -0.0136130746 -0.0151020048 
##           145           146           147           148           149 
## -0.6491330093  0.5089412804  0.0198081074 -0.2184427088 -0.0626808175 
##           150           151           152           153           154 
## -0.0232161815 -0.0247051117  0.3079386518 -0.6193300126 -0.0351904552 
##           155           156           157           158           159 
## -0.0414229027  0.0188372528 -0.0307096957 -0.0321986260 -0.0149869349 
##           160           161           162           163           164 
## -0.0036051011 -0.0757256960 -0.0116401080 -0.1742037359 -0.0357920263 
##           165           166           167           168           169 
## -0.0372032920 -0.2483563235 -0.2813997227 -0.1067490983 -0.0787267436 
##           170           171           172           173           174 
##  0.0786696808 -0.0471878676 -0.0486563388  0.2077356296  0.0019712878 
##           175           176           177           178           179 
## -0.0805729466  0.0865198432  0.0129930756 -0.0336060184 -0.0351717909 
##           180           181           182           183           184 
## -0.5430052770  0.5699702406 -0.0347356633 -0.0104205063 -0.0060264274 
##           185           186           187           188           189 
## -0.0281139827 -0.0296381870  0.7002595730 -0.1527768573  0.1775209363 
##           190           191           192           193           194 
##  0.1674719834  0.1528291914  0.0072232150  0.0059096600  0.2073372355 
##           195           196           197           198           199 
##  0.0335530920 -0.1284625207  0.0521549363  0.1490383312  0.0134896319 
##           200           201           202           203           204 
##  0.0124702152  0.2319947813  0.1609154434 -0.1409760511  0.1574839305 
##           205           206           207           208           209 
##  0.0712239389  0.0229084624  0.0220056565  0.1333588061  0.2773868323 
##           210           211           212           213           214 
## -0.4082967438 -0.0289206932 -0.0188004247  0.0180399290  0.0166556427 
##           215           216           217           218           219 
##  0.1653146892 -0.2047006295  0.1675126483  0.0762783587  0.0738031871 
##           220           221           222           223           224 
##  0.0143720963  0.0128985843  0.0468493669  0.0396119258 -0.0621725083 
##           225           226           227           228           229 
##  0.0147808954  0.0266446386  0.0170979071  0.0158166921  0.1304838681 
##           230           231           232           233           234 
##  0.0297638686 -0.1115102331  0.0400706930  0.0234368102  0.0175708037 
##           235           236           237           238           239 
##  0.0161697623  0.0459638561 -0.0108645017 -0.0137342257 -0.0215486015 
##           240           241           242           243           244 
## -0.0174015080  0.0122976291  0.0108086989 -0.5462422658  0.4987813802 
##           245           246           247           248           249 
## -0.0827648429 -0.0078687278 -0.0051523814  0.0069010163  0.0054120861 
##           250           251           252           253           254 
##  0.5476827967 -0.3030223961 -0.0066957590  0.0184288810  0.1092836405 
##           255           256           257           258           259 
##  0.0172406784  0.0157517482 -0.2118031721 -0.0219490663 -0.0494768935 
##           260           261           262           263           264 
## -0.0182412713 -0.0355656984  0.0047338360  0.0032449058 -0.0303316544 
##           265           266           267           268           269 
## -0.0461200114 -0.0402549344  0.0040586207 -0.0396813616 -0.0003193617 
##           270           271           272           273           274 
## -0.0018082919 -0.0109040051 -0.0518081474 -0.0831792400  0.0242242065 
##           275           276           277           278           279 
## -0.0054861990 -0.0037181888 -0.0052071191 -0.0785204405 -0.0515469270 
##           280           281           282           283           284 
## -0.0901820582 -0.0732495831 -0.0398027942 -0.0131189694 -0.0146078997 
##           285           286           287           288           289 
## -0.1030378317 -0.0585937511 -0.0740603660 -0.0188162420 -0.0889480916 
##           290           291           292           293           294 
## -0.0213484116 -0.0228373419 -0.0009888652 -0.0583160775 -0.0933343153 
##           295           296           297           298           299 
## -0.2939240825  0.2093383182 -0.0247814122 -0.0262703424 -0.1015690145 
##           300           301           302           303           304 
## -0.0390658512 -0.1067096847  0.1278524620 -0.2962888130 -0.0337768716 
##           305           306           307           308           309 
## -0.0352658018  0.0648706161 -0.0386313636 -0.0412756075 -0.2067728246 
##           310           311           312           313           314 
##  0.2027474944 -0.0284536161 -0.0299425463 -0.1194382612 -0.0796777732 
##           315           316           317           318           319 
## -0.0893594924  0.1835312754 -0.2090382615 -0.0333254430 -0.0348143732 
##           320           321           322           323           324 
##  0.0695995113 -0.1067974520 -0.1118620619 -0.2982037909 -0.3142264635 
##           325           326           327           328           329 
## -0.0526601978 -0.0541491280  0.2644388950 -0.0093278635  0.0558597065 
##           330           331           332           333           334 
##  0.2707293092  0.1847381877 -0.0195737102 -0.0210626404  0.1126526669 
##           335           336           337           338           339 
##  0.0332107496  0.0273795097  0.0511426205  0.0341275014 -0.0080152001 
##           340           341           342           343           344 
## -0.0095041303 -0.0069758511 -0.0526065691 -0.0957398444 -0.0090761737 
##           345           346           347           348           349 
##  0.0160888997 -0.0108625075 -0.0123514377 -0.0010008676 -0.0228389005 
##           350           351           352           353           354 
## -0.0472800585 -0.2545202184 -0.2846204474 -0.0288026761 -0.0302916063 
##           355           356           357           358           359 
## -0.9058903735  1.5652411290 -0.1628820768  0.3315450181  0.0155505485 
##           360           361           362           363           364 
##  0.0034116287  0.0019226985  0.7549825649 -0.2684037661  0.2725795109 
##           365           366           367           368           369 
##  0.1214239086  0.2670232421  0.0389337198  0.0374447896  0.1721576193 
##           370           371           372           373           374 
##  0.0861561917  0.0188683809  0.1833389820  0.1157649150  0.0514240222 
##           375           376           377           378           379 
##  0.0499350920 -0.3315783638  0.6859096507  0.0254283685  0.0111584290 
##           380           381           382           383           384 
## -0.0978447114  0.0527690387  0.0512801085  0.6068872423 -0.3934194425 
##           385           386           387           388           389 
## -0.0123567000  0.1404074447  0.2065508315  0.0566622114  0.0555163791 
##           390           391           392           393           394 
## -0.2125527694  0.0638413436 -0.2490112019 -0.1070191973 -0.0657752020 
##           395           396           397           398           399 
##  0.0397380690  0.0381994992 -0.3181207230 -0.1465143113  0.1600813958 
##           400           401           402           403           404 
## -0.1376118862 -0.0233191256  0.0146532435  0.0125864130 -0.0450035769 
##           405           406           407           408           409 
## -0.0729973980  0.1954139556  0.0725111900 -0.0485840740  0.0192594298 
##           410           411           412           413           414 
##  0.0173593428 -0.0300150667 -0.1064081660 -0.0606890359 -0.1471772732 
##           415           416           417           418           419 
## -0.0472908167  0.0136989670  0.0117901025 -0.3343997815 -0.0541140700 
##           420           421           422           423           424 
##  0.0739029076 -0.0423713434 -0.0573320347  0.0012472156 -0.0006758919 
##           425           426           427           428           429 
##  0.0103248355 -0.0569044785  0.1365386768 -0.0364978247  0.0830236429 
##           430           431           432           433           434 
##  0.0121966072  0.0103500213 -0.0807190521 -0.1064033790  0.0816416569 
##           435           436           437           438           439 
## -0.0374279359 -0.0177343901  0.0083650951  0.0066315216  0.1274158456 
##           440           441           442           443           444 
##  0.0340155770  0.1250497496  0.1538768441  0.1140251474  0.0283837841 
##           445           446           447           448           449 
##  0.0267832477  0.1310663625 -0.0160997119  0.0922018757  0.0438603288 
##           450           451           452           453           454 
## -0.0559222034  0.0299940801  0.0285615872 -0.0548322644 -0.0089794018 
##           455           456           457           458           459 
## -0.2664375111 -0.3134873862 -0.3155900116 -0.0083381840 -0.0094403393 
##           460           461           462           463           464 
## -0.8998684248 -0.9099201774  1.8229177938 -0.0278178361 -0.0187554169 
##           465           466           467           468           469 
## -0.0060663639 -0.0070574181 -0.0236574378  0.0089609705 -0.1985607999 
##           470           471           472           473           474 
## -0.0516989335 -0.0506748450 -0.0188521589 -0.0200424954 -0.0440537621 
##           475           476           477           478           479 
## -0.0327497766  0.4277819064 -0.0278734106 -0.0275121563 -0.0096476569 
##           480           481           482           483           484 
## -0.0110513494 -0.0102416409 -0.0353550298 -0.0551953313 -0.0147304285 
##           485           486           487           488           489 
## -0.0113245502 -0.0152254509 -0.0166016525  0.0249369224 -0.0137193043 
##           490 
## -0.2448440774

mod_ardl114 <- ardlDlm(log_cases ~ log_viral,
                       data = nc_data_train,p=14,q=1)
summary(mod_ardl114)
## 
## Time series regression with "ts" data:
## Start = 15, End = 490
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.90992 -0.04728 -0.00450  0.03391  1.82292 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.0510420  0.0547900  -0.932  0.35204    
## log_viral.t   0.0068198  0.0962389   0.071  0.94354    
## log_viral.1  -0.0004429  0.1355423  -0.003  0.99739    
## log_viral.2  -0.0662456  0.1355423  -0.489  0.62526    
## log_viral.3  -0.0267071  0.1355444  -0.197  0.84389    
## log_viral.4   0.3559041  0.1355462   2.626  0.00894 ** 
## log_viral.5  -0.2269657  0.1359995  -1.669  0.09582 .  
## log_viral.6  -0.0071780  0.1363630  -0.053  0.95804    
## log_viral.7  -0.0289224  0.1364897  -0.212  0.83228    
## log_viral.8  -0.0003262  0.1361771  -0.002  0.99809    
## log_viral.9   0.1139565  0.1361771   0.837  0.40313    
## log_viral.10 -0.0872283  0.1361834  -0.641  0.52215    
## log_viral.11  0.0869370  0.1361774   0.638  0.52353    
## log_viral.12 -0.0677954  0.1362716  -0.498  0.61907    
## log_viral.13  0.0133665  0.1363593   0.098  0.92196    
## log_viral.14 -0.0393125  0.0967766  -0.406  0.68477    
## log_cases.1   0.9662545  0.0113959  84.790  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1981 on 459 degrees of freedom
## Multiple R-squared:  0.9661, Adjusted R-squared:  0.9649 
## F-statistic: 818.1 on 16 and 459 DF,  p-value: < 2.2e-16
f_ardl114  <- forecast(mod_ardl114, 
                       x= t(nc_data_test[,5]),
                       h=14,
                       interval = TRUE)
rmse(nc_data_test$log_cases,
     f_ardl114$forecasts[,2])
## [1] 0.1527168
mae(nc_data_test$log_cases,
     f_ardl114$forecasts[,2])
## [1] 0.1284164
checkresiduals(mod_ardl114)
## Time Series:
## Start = 15 
## End = 490 
## Frequency = 1 
##            15            16            17            18            19 
## -0.0079361204  0.0749551140  0.0104102776  0.0417587039  0.1788979741 
##            20            21            22            23            24 
## -0.0197832873 -0.0636556796  0.0659424356 -0.0058961325  0.0563267039 
##            25            26            27            28            29 
##  0.0722532860  0.0555601394  0.0195105611  0.1191322887  0.0569816684 
##            30            31            32            33            34 
##  0.0432000544  0.0106782259  0.0277613061  0.0608521145  0.0164457056 
##            35            36            37            38            39 
## -0.1385073028  0.0055273619  0.0465037029  0.0195600474 -0.0073049436 
##            40            41            42            43            44 
##  0.0448363228  0.0049317851 -0.0320415286  0.0338699326  0.0305961503 
##            45            46            47            48            49 
##  0.0283465099  0.0926974719  0.0618128861 -0.0184534334  0.0213206569 
##            50            51            52            53            54 
##  0.0326426656 -0.0040375544 -0.0371331073 -0.1782426977  0.2204057225 
##            55            56            57            58            59 
##  0.0104794007 -0.0693380908 -0.0828500659  0.0462878989  0.0246169661 
##            60            61            62            63            64 
##  0.0262632746 -0.0843961025 -0.0379274320 -0.0720000488  0.0817978511 
##            65            66            67            68            69 
## -0.0246907811 -0.1745607217  0.0117215188  0.2851422174  0.0443775771 
##            70            71            72            73            74 
##  0.0028641604 -0.0653787188  0.0414945238  0.1727227919  0.0492103829 
##            75            76            77            78            79 
## -0.1762951948 -0.0672963539 -0.0814167277  0.1228827473  0.0513663051 
##            80            81            82            83            84 
## -0.1715105563 -0.0272278816  0.3267267423  0.0269302279 -0.0049594610 
##            85            86            87            88            89 
## -0.0068720516 -0.1771227030  0.0139272167  0.0124382865  0.0116317455 
##            90            91            92            93            94 
## -0.0586408875 -0.0057825049 -0.0022346363  0.2057315708  0.0174397517 
##            95            96            97            98            99 
##  0.0159508215  0.0837310328  0.0630946687 -0.0019086285  0.0872437669 
##           100           101           102           103           104 
##  0.0692130295  0.0246579464  0.0231690162  0.0438558048 -0.0308327461 
##           105           106           107           108           109 
## -0.0682791666  0.0288945455  0.0165520992  0.0209345474  0.0194456172 
##           110           111           112           113           114 
##  0.0187354925  0.0234234992  0.0130892478  0.0212016080  0.0359539623 
##           115           116           117           118           119 
##  0.0216787007  0.0201897705 -0.0124944339  0.0329172523 -0.0371419702 
##           120           121           122           123           124 
## -0.0537262902  0.0013843828  0.0164835277  0.0149945975  0.0220603837 
##           125           126           127           128           129 
## -0.1017259910  0.0188264098 -0.0081172809 -0.0540654425  0.0104045561 
##           130           131           132           133           134 
##  0.0089156258 -0.0880293066  0.0128322586 -0.1653236580  0.0155740275 
##           135           136           137           138           139 
## -0.0111191180  0.0019533161  0.0004643858 -0.1051715333 -0.0873028623 
##           140           141           142           143           144 
## -0.1879690028 -0.0661025606 -0.0561235139 -0.0136130746 -0.0151020048 
##           145           146           147           148           149 
## -0.6491330093  0.5089412804  0.0198081074 -0.2184427088 -0.0626808175 
##           150           151           152           153           154 
## -0.0232161815 -0.0247051117  0.3079386518 -0.6193300126 -0.0351904552 
##           155           156           157           158           159 
## -0.0414229027  0.0188372528 -0.0307096957 -0.0321986260 -0.0149869349 
##           160           161           162           163           164 
## -0.0036051011 -0.0757256960 -0.0116401080 -0.1742037359 -0.0357920263 
##           165           166           167           168           169 
## -0.0372032920 -0.2483563235 -0.2813997227 -0.1067490983 -0.0787267436 
##           170           171           172           173           174 
##  0.0786696808 -0.0471878676 -0.0486563388  0.2077356296  0.0019712878 
##           175           176           177           178           179 
## -0.0805729466  0.0865198432  0.0129930756 -0.0336060184 -0.0351717909 
##           180           181           182           183           184 
## -0.5430052770  0.5699702406 -0.0347356633 -0.0104205063 -0.0060264274 
##           185           186           187           188           189 
## -0.0281139827 -0.0296381870  0.7002595730 -0.1527768573  0.1775209363 
##           190           191           192           193           194 
##  0.1674719834  0.1528291914  0.0072232150  0.0059096600  0.2073372355 
##           195           196           197           198           199 
##  0.0335530920 -0.1284625207  0.0521549363  0.1490383312  0.0134896319 
##           200           201           202           203           204 
##  0.0124702152  0.2319947813  0.1609154434 -0.1409760511  0.1574839305 
##           205           206           207           208           209 
##  0.0712239389  0.0229084624  0.0220056565  0.1333588061  0.2773868323 
##           210           211           212           213           214 
## -0.4082967438 -0.0289206932 -0.0188004247  0.0180399290  0.0166556427 
##           215           216           217           218           219 
##  0.1653146892 -0.2047006295  0.1675126483  0.0762783587  0.0738031871 
##           220           221           222           223           224 
##  0.0143720963  0.0128985843  0.0468493669  0.0396119258 -0.0621725083 
##           225           226           227           228           229 
##  0.0147808954  0.0266446386  0.0170979071  0.0158166921  0.1304838681 
##           230           231           232           233           234 
##  0.0297638686 -0.1115102331  0.0400706930  0.0234368102  0.0175708037 
##           235           236           237           238           239 
##  0.0161697623  0.0459638561 -0.0108645017 -0.0137342257 -0.0215486015 
##           240           241           242           243           244 
## -0.0174015080  0.0122976291  0.0108086989 -0.5462422658  0.4987813802 
##           245           246           247           248           249 
## -0.0827648429 -0.0078687278 -0.0051523814  0.0069010163  0.0054120861 
##           250           251           252           253           254 
##  0.5476827967 -0.3030223961 -0.0066957590  0.0184288810  0.1092836405 
##           255           256           257           258           259 
##  0.0172406784  0.0157517482 -0.2118031721 -0.0219490663 -0.0494768935 
##           260           261           262           263           264 
## -0.0182412713 -0.0355656984  0.0047338360  0.0032449058 -0.0303316544 
##           265           266           267           268           269 
## -0.0461200114 -0.0402549344  0.0040586207 -0.0396813616 -0.0003193617 
##           270           271           272           273           274 
## -0.0018082919 -0.0109040051 -0.0518081474 -0.0831792400  0.0242242065 
##           275           276           277           278           279 
## -0.0054861990 -0.0037181888 -0.0052071191 -0.0785204405 -0.0515469270 
##           280           281           282           283           284 
## -0.0901820582 -0.0732495831 -0.0398027942 -0.0131189694 -0.0146078997 
##           285           286           287           288           289 
## -0.1030378317 -0.0585937511 -0.0740603660 -0.0188162420 -0.0889480916 
##           290           291           292           293           294 
## -0.0213484116 -0.0228373419 -0.0009888652 -0.0583160775 -0.0933343153 
##           295           296           297           298           299 
## -0.2939240825  0.2093383182 -0.0247814122 -0.0262703424 -0.1015690145 
##           300           301           302           303           304 
## -0.0390658512 -0.1067096847  0.1278524620 -0.2962888130 -0.0337768716 
##           305           306           307           308           309 
## -0.0352658018  0.0648706161 -0.0386313636 -0.0412756075 -0.2067728246 
##           310           311           312           313           314 
##  0.2027474944 -0.0284536161 -0.0299425463 -0.1194382612 -0.0796777732 
##           315           316           317           318           319 
## -0.0893594924  0.1835312754 -0.2090382615 -0.0333254430 -0.0348143732 
##           320           321           322           323           324 
##  0.0695995113 -0.1067974520 -0.1118620619 -0.2982037909 -0.3142264635 
##           325           326           327           328           329 
## -0.0526601978 -0.0541491280  0.2644388950 -0.0093278635  0.0558597065 
##           330           331           332           333           334 
##  0.2707293092  0.1847381877 -0.0195737102 -0.0210626404  0.1126526669 
##           335           336           337           338           339 
##  0.0332107496  0.0273795097  0.0511426205  0.0341275014 -0.0080152001 
##           340           341           342           343           344 
## -0.0095041303 -0.0069758511 -0.0526065691 -0.0957398444 -0.0090761737 
##           345           346           347           348           349 
##  0.0160888997 -0.0108625075 -0.0123514377 -0.0010008676 -0.0228389005 
##           350           351           352           353           354 
## -0.0472800585 -0.2545202184 -0.2846204474 -0.0288026761 -0.0302916063 
##           355           356           357           358           359 
## -0.9058903735  1.5652411290 -0.1628820768  0.3315450181  0.0155505485 
##           360           361           362           363           364 
##  0.0034116287  0.0019226985  0.7549825649 -0.2684037661  0.2725795109 
##           365           366           367           368           369 
##  0.1214239086  0.2670232421  0.0389337198  0.0374447896  0.1721576193 
##           370           371           372           373           374 
##  0.0861561917  0.0188683809  0.1833389820  0.1157649150  0.0514240222 
##           375           376           377           378           379 
##  0.0499350920 -0.3315783638  0.6859096507  0.0254283685  0.0111584290 
##           380           381           382           383           384 
## -0.0978447114  0.0527690387  0.0512801085  0.6068872423 -0.3934194425 
##           385           386           387           388           389 
## -0.0123567000  0.1404074447  0.2065508315  0.0566622114  0.0555163791 
##           390           391           392           393           394 
## -0.2125527694  0.0638413436 -0.2490112019 -0.1070191973 -0.0657752020 
##           395           396           397           398           399 
##  0.0397380690  0.0381994992 -0.3181207230 -0.1465143113  0.1600813958 
##           400           401           402           403           404 
## -0.1376118862 -0.0233191256  0.0146532435  0.0125864130 -0.0450035769 
##           405           406           407           408           409 
## -0.0729973980  0.1954139556  0.0725111900 -0.0485840740  0.0192594298 
##           410           411           412           413           414 
##  0.0173593428 -0.0300150667 -0.1064081660 -0.0606890359 -0.1471772732 
##           415           416           417           418           419 
## -0.0472908167  0.0136989670  0.0117901025 -0.3343997815 -0.0541140700 
##           420           421           422           423           424 
##  0.0739029076 -0.0423713434 -0.0573320347  0.0012472156 -0.0006758919 
##           425           426           427           428           429 
##  0.0103248355 -0.0569044785  0.1365386768 -0.0364978247  0.0830236429 
##           430           431           432           433           434 
##  0.0121966072  0.0103500213 -0.0807190521 -0.1064033790  0.0816416569 
##           435           436           437           438           439 
## -0.0374279359 -0.0177343901  0.0083650951  0.0066315216  0.1274158456 
##           440           441           442           443           444 
##  0.0340155770  0.1250497496  0.1538768441  0.1140251474  0.0283837841 
##           445           446           447           448           449 
##  0.0267832477  0.1310663625 -0.0160997119  0.0922018757  0.0438603288 
##           450           451           452           453           454 
## -0.0559222034  0.0299940801  0.0285615872 -0.0548322644 -0.0089794018 
##           455           456           457           458           459 
## -0.2664375111 -0.3134873862 -0.3155900116 -0.0083381840 -0.0094403393 
##           460           461           462           463           464 
## -0.8998684248 -0.9099201774  1.8229177938 -0.0278178361 -0.0187554169 
##           465           466           467           468           469 
## -0.0060663639 -0.0070574181 -0.0236574378  0.0089609705 -0.1985607999 
##           470           471           472           473           474 
## -0.0516989335 -0.0506748450 -0.0188521589 -0.0200424954 -0.0440537621 
##           475           476           477           478           479 
## -0.0327497766  0.4277819064 -0.0278734106 -0.0275121563 -0.0096476569 
##           480           481           482           483           484 
## -0.0110513494 -0.0102416409 -0.0353550298 -0.0551953313 -0.0147304285 
##           485           486           487           488           489 
## -0.0113245502 -0.0152254509 -0.0166016525  0.0249369224 -0.0137193043 
##           490 
## -0.2448440774

nc_data_test <- cbind(nc_data_test,f_ardl114$forecasts[,2],f_ardl114$forecasts[,1],
                      f_ardl114$forecasts[,3])

png(filename = "sensitivity_nc_ardl.png",res = 700, units = "cm",
    width = 20, height = 10)
ardl_plot <- nc_data_train %>% ggplot(aes(date,log_cases)) + 
  geom_line() + 
  geom_ribbon(data = nc_data_test , aes(ymin = f_ardl114$forecasts[,1], 
                                        ymax = f_ardl114$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) +
  geom_line(data = nc_data_test,aes(date,log_cases,color="Actual")) +
  geom_line(data = nc_data_test,aes(date,f_ardl114$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "bottom") + ylab("")
dev.off()
## quartz_off_screen 
##                 2
exp(f_ardl114 $forecasts[1,2])
## [1] 11.95761
exp(f_ardl114 $forecasts[1,1])
## [1] 8.101852
exp(f_ardl114 $forecasts[1,3])
## [1] 17.72089
exp(f_ardl114 $forecasts[1,2]) - exp(nc_data_test[1,4])
## [1] 0.2743545
exp(f_ardl114 $forecasts[7,2])
## [1] 14.73769
exp(f_ardl114 $forecasts[7,1])
## [1] 5.738535
exp(f_ardl114 $forecasts[7,3])
## [1] 36.01211
exp(f_ardl114$forecasts[7,2]) - exp(nc_data_test[7,4])
## [1] -3.410547
exp(f_ardl114 $forecasts[14,2])
## [1] 17.70536
exp(f_ardl114$forecasts[14,1])
## [1] 5.575139
exp(f_ardl114$forecasts[14,3])
## [1] 56.09214
exp(f_ardl114 $forecasts[14,2]) - exp(nc_data_test[14,4])
## [1] -7.62401
#Distributed lag model

lowest_rmse_dl <- Inf
best_mod_dl <- NULL

for (q in seq(1,14)){
  mod <- dlm(log_cases ~ log_viral,
             data = nc_data_train,q=q)
  f <- forecast(mod, x= t(nc_data_test[,5]),h=14)
  forecast_acc <- mae(nc_data_test[,4],
                      f$forecasts)
  if (forecast_acc<lowest_rmse_dl){
    lowest_rmse_dl <- forecast_acc
    best_mod_dl <-mod 
  }
}


lowest_rmse_dl #0.622
## [1] 0.5919259
summary(best_mod_dl) #DL(14)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.29091 -0.53312  0.09167  0.43047  2.03724 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.7698170  0.2207213  -3.488 0.000534 ***
## log_viral.t   0.0708392  0.3924113   0.181 0.856822    
## log_viral.1   0.0005531  0.5526871   0.001 0.999202    
## log_viral.2  -0.0656752  0.5526871  -0.119 0.905463    
## log_viral.3  -0.0902575  0.5526871  -0.163 0.870349    
## log_viral.4   0.2687696  0.5526871   0.486 0.626989    
## log_viral.5   0.0715636  0.5543655   0.129 0.897342    
## log_viral.6   0.0233671  0.5560317   0.042 0.966497    
## log_viral.7  -0.1222001  0.5565320  -0.220 0.826300    
## log_viral.8   0.0005348  0.5552756   0.001 0.999232    
## log_viral.9   0.1145059  0.5552756   0.206 0.836715    
## log_viral.10  0.0233234  0.5552756   0.042 0.966514    
## log_viral.11  0.1095497  0.5552756   0.197 0.843689    
## log_viral.12  0.0145768  0.5556466   0.026 0.979082    
## log_viral.13  0.0508206  0.5560157   0.091 0.927214    
## log_viral.14  0.1952651  0.3944549   0.495 0.620819    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8076 on 460 degrees of freedom
## Multiple R-squared:  0.4355, Adjusted R-squared:  0.4171 
## F-statistic: 23.66 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##       AIC      BIC
## 1 1165.17 1235.982
checkresiduals(best_mod_dl)
##            1            2            3            4            5            6 
##  1.536593406  1.551576009  1.503886164  1.486720187  1.607319380  1.525131019 
##            7            8            9           10           11           12 
##  1.401871490  1.412649754  1.350963648  1.355959990  1.374280435  1.375336126 
##           13           14           15           16           17           18 
##  1.340270000  1.406036663  1.407713604  1.395290258  1.353142299  1.327065848 
##           19           20           21           22           23           24 
##  1.335006793  1.298236744  1.107782248  1.068069622  1.070411373  1.048108430 
##           25           26           27           28           29           30 
##  0.997259244  1.000313879  0.963324283  0.890637343  0.886594946  0.879153095 
##           31           32           33           34           35           36 
##  0.872090716  0.927183748  0.949579678  0.890916913  0.874035580  0.869326150 
##           37           38           39           40           41           42 
##  0.827833339  0.757023153  0.545059025  0.738942774  0.716320886  0.614672632 
##           43           44           45           46           47           48 
##  0.503222798  0.524409773  0.525588834  0.525940541  0.415667627  0.355547982 
##           49           50           51           52           53           54 
##  0.263412225  0.328463759  0.284569382  0.094664284  0.095015991  0.368823153 
##           55           56           57           58           59           60 
##  0.392589306  0.374067781  0.288208620  0.311857977  0.468315526  0.493547057 
##           61           62           63           64           65           66 
##  0.292468184  0.207137041  0.110592804  0.221886199  0.257645419  0.071699049 
##           67           68           69           70           71           72 
##  0.033876328  0.351331202  0.358240280  0.333054258  0.307085786  0.111480896 
##           73           74           75           76           77           78 
##  0.115904691  0.116256398  0.115836320  0.045121171  0.029678459  0.018584964 
##           79           80           81           82           83           84 
##  0.215569948  0.219993742  0.220345449  0.288512124  0.333705503  0.312398251 
##           85           86           87           88           89           90 
##  0.381242644  0.429471028  0.433894823  0.434246530  0.455319782  0.400956742 
##           91           92           93           94           95           96 
##  0.311009526  0.321551561  0.319133319  0.323557114  0.323908821  0.323585160 
##           97           98           99          100          101          102 
##  0.327923813  0.321809543  0.324294188  0.341185259  0.345609054  0.345960760 
##          103          104          105          106          107          108 
##  0.313663019  0.327830253  0.271487922  0.200742797  0.187233589  0.191657384 
##          109          110          111          112          113          114 
##  0.192009090  0.199461340  0.082839121  0.090732513  0.071696075  0.007091785 
##          115          116          117          118          119          120 
##  0.011515580  0.011867287 -0.084691182 -0.077166288 -0.248023504 -0.231937149 
##          121          122          123          124          125          126 
## -0.243348866 -0.238925071 -0.238573364 -0.343822820 -0.427688526 -0.609362545 
##          127          128          129          130          131          132 
## -0.662759219 -0.704637034 -0.700213239 -0.699861532 -1.333506074 -0.787730300 
##          133          134          135          136          137          138 
## -0.749477427 -0.950486005 -0.989211645 -0.984787850 -0.984436143 -0.651405916 
##          139          140          141          142          143          144 
## -1.256919233 -1.257831917 -1.264666022 -1.211271437 -1.206847642 -1.206495935 
##          145          146          147          148          149          150 
## -1.188897781 -1.160548265 -1.205248274 -1.184074045 -1.326440063 -1.334438506 
##          151          152          153          154          155          156 
## -1.334183795 -1.545050378 -1.781865161 -1.836023690 -1.866861091 -1.732754075 
##          157          158          159          160          161          162 
## -1.701025221 -1.700721409 -1.443992009 -1.401718794 -1.443395606 -1.308954457 
##          163          164          165          166          167          168 
## -1.260139179 -1.292130475 -1.291672229 -1.799009565 -1.176305402 -1.179277836 
##          169          170          171          172          173          174 
## -1.155137562 -1.130080806 -1.123418340 -1.123008256 -0.392664066 -0.540048580 
##          175          176          177          178          179          180 
## -0.352125651 -0.183074443 -0.031884634 -0.033014202 -0.032881923  0.168705864 
##          181          182          183          184          185          186 
##  0.189706464  0.047979883  0.076721592  0.216238175  0.213594953  0.213310121 
##          187          188          189          190          191          192 
##  0.432565593  0.573412077  0.407552322  0.525367395  0.573217045  0.541321379 
##          193          194          195          196          197          198 
##  0.540842936  0.651728114  0.903001334  0.460022129  0.400370189  0.363782137 
##          199          200          201          202          203          204 
##  0.332934265  0.333058933  0.481871860  0.255649960  0.409269991  0.496957823 
##          205          206          207          208          209          210 
##  0.548900285  0.347292287  0.347665591  0.382024461  0.407945670  0.331239052 
##          211          212          213          214          215          216 
##  0.316304987  0.331429143  0.412483890  0.412541656  0.527292753  0.537463943 
##          217          218          219          220          221          222 
##  0.406004024  0.426740914  0.433951963  0.418395655  0.418603265  0.448635783 
##          223          224          225          226          227          228 
##  0.420814155  0.391068798  0.361120007  0.329769986  0.281579274  0.281930981 
##          229          230          231          232          233          234 
## -0.274733520  0.232373921  0.140850293  0.127591516  0.117234424  0.121658219 
##          235          236          237          238          239          240 
##  0.122009926  0.664667100  0.338270240  0.319242174  0.326261086  0.423635816 
##          241          242          243          244          245          246 
##  0.428059611  0.428411318  0.201242861  0.171557804  0.115374395  0.092602770 
##          247          248          249          250          251          252 
##  0.053013074  0.057436869  0.057788576  0.024598478 -0.023296574 -0.063682570 
##          253          254          255          256          257          258 
## -0.058111939 -0.096731358 -0.092307563 -0.091955857 -0.100665107 -0.150021215 
##          259          260          261          262          263          264 
## -0.229055132 -0.197738339 -0.197450836 -0.193027041 -0.192675335 -0.265602193 
##          265          266          267          268          269          270 
## -0.309131200 -0.389798693 -0.450531321 -0.476029792 -0.471605997 -0.471254290 
##          271          272          273          274          275          276 
## -0.559297759 -0.599962693 -0.654694245 -0.652054504 -0.719897776 -0.715473982 
##          277          278          279          280          281          282 
## -0.715122275 -0.692887335 -0.728766550 -0.798425504 -1.066043323 -0.821629934 
##          283          284          285          286          287          288 
## -0.817206140 -0.816854433 -0.891766642 -0.901684353 -0.978883482 -0.818635115 
##          289          290          291          292          293          294 
## -1.088197766 -1.083773972 -1.083422265 -0.982899384 -0.989307289 -0.998115463 
##          295          296          297          298          299          300 
## -1.171843391 -0.930450552 -0.926026757 -0.925675050 -1.014784302 -1.061162645 
##          301          302          303          304          305          306 
## -1.115629910 -0.895088156 -1.074820311 -1.070396516 -1.070044810 -0.965244462 
##          307          308          309          310          311          312 
## -1.040414230 -1.118084229 -1.379194720 -1.647778669 -1.643354874 -1.643003167 
##          313          314          315          316          317          318 
## -1.324028681 -1.289621514 -1.191160126 -0.880871535 -0.667306988 -0.662883193 
##          319          320          321          322          323          324 
## -0.662531486 -0.528429716 -0.478331805 -0.435727974 -0.370518493 -0.324786740 
##          325          326          327          328          329          330 
## -0.320362945 -0.320011239 -0.317096496 -0.359947445 -0.444457906 -0.439172623 
##          331          332          333          334          335          336 
## -0.409162705 -0.404738910 -0.404387203 -0.392650170 -0.403183856 -0.437775497 
##          337          338          339          340          341          342 
## -0.678159760 -0.940794454 -0.936370659 -0.936018952 -1.811231256 -0.185814212 
##          343          344          345          346          347          348 
## -0.343343114 -0.000848807  0.013831312  0.018255107  0.018606814  0.772053143 
##          349          350          351          352          353          354 
##  0.476651120  0.732228594  0.828306109  1.066478690  1.070902485  1.071254192 
##          355          356          357          358          359          360 
##  1.206353485  1.250855746  1.226596184  1.367906100  1.436611294  1.441035089 
##          361          362          363          364          365          366 
##  1.441386795  1.060259803  1.709445524  1.676270617  1.630225501  1.476468974 
##          367          368          369          370          371          372 
##  1.480892769  1.481244476  2.037238073  1.574126100  1.507732544  1.596623841 
##          373          374          375          376          377          378 
##  1.748396762  1.697942945  1.697866152  1.429741633  1.447093373  1.150974958 
##          379          380          381          382          383          384 
##  0.977041114  0.879864019  1.027309129  1.027624127  0.671654076  0.499273671 
##          385          386          387          388          389          390 
##  0.639327250  0.521937025  0.478084054  0.304646360  0.305806787  0.249435775 
##          391          392          393          394          395          396 
##  0.166801788  0.355509520  0.430344089  0.366283556  0.429895045  0.430862451 
##          397          398          399          400          401          402 
##  0.384507750  0.263184454  0.191789518  0.030870869 -0.019281431  0.150064647 
##          403          404          405          406          407          408 
##  0.150983831 -0.194235131 -0.247649753 -0.171137346 -0.193966235 -0.250361891 
##          409          410          411          412          413          414 
## -0.232545257 -0.231562172 -0.219525362 -0.275266216 -0.135565791 -0.171699933 
##          415          416          417          418          419          420 
## -0.088964302  0.028150138  0.029007791 -0.061154063 -0.174078710 -0.095048432 
##          421          422          423          424          425          426 
## -0.130341489 -0.152091940 -0.098790597 -0.098074107  0.023471879  0.047418264 
##          427          428          429          430          431          432 
##  0.161670229  0.298407458  0.393178738  0.469642037  0.470159878  0.575000338 
##          433          434          435          436          437          438 
##  0.529072399  0.593047211  0.605653855  0.518937020  0.538345220  0.538638850 
##          439          440          441          442          443          444 
##  0.455571381  0.421132727  0.130417922 -0.207275816 -0.525974878 -0.529672662 
##          445          446          447          448          449          450 
## -0.529829378 -1.420394844 -2.290911786 -0.399258326 -0.445433336 -0.457853386 
##          451          452          453          454          455          456 
## -0.445247845 -0.445610524 -0.462559236 -0.446234905 -0.638056496 -0.682071998 
##          457          458          459          460          461          462 
## -0.718082998 -0.800597143 -0.800693401 -0.824778925 -0.836709246 -0.387741136 
##          463          464          465          466          467          468 
## -0.404803053 -0.425674312 -0.507977759 -0.507763137 -0.506707833 -0.530815066 
##          469          470          471          472          473          474 
## -0.573940919 -0.569819727 -0.567717078 -0.625579768 -0.625370813 -0.583592938 
##          475          476 
## -0.581894113 -0.811369602

mod_dl14 <- dlm(log_cases ~ log_viral,
               data = nc_data_train,q=14)
summary(mod_dl14)
## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.29091 -0.53312  0.09167  0.43047  2.03724 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.7698170  0.2207213  -3.488 0.000534 ***
## log_viral.t   0.0708392  0.3924113   0.181 0.856822    
## log_viral.1   0.0005531  0.5526871   0.001 0.999202    
## log_viral.2  -0.0656752  0.5526871  -0.119 0.905463    
## log_viral.3  -0.0902575  0.5526871  -0.163 0.870349    
## log_viral.4   0.2687696  0.5526871   0.486 0.626989    
## log_viral.5   0.0715636  0.5543655   0.129 0.897342    
## log_viral.6   0.0233671  0.5560317   0.042 0.966497    
## log_viral.7  -0.1222001  0.5565320  -0.220 0.826300    
## log_viral.8   0.0005348  0.5552756   0.001 0.999232    
## log_viral.9   0.1145059  0.5552756   0.206 0.836715    
## log_viral.10  0.0233234  0.5552756   0.042 0.966514    
## log_viral.11  0.1095497  0.5552756   0.197 0.843689    
## log_viral.12  0.0145768  0.5556466   0.026 0.979082    
## log_viral.13  0.0508206  0.5560157   0.091 0.927214    
## log_viral.14  0.1952651  0.3944549   0.495 0.620819    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8076 on 460 degrees of freedom
## Multiple R-squared:  0.4355, Adjusted R-squared:  0.4171 
## F-statistic: 23.66 on 15 and 460 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##       AIC      BIC
## 1 1165.17 1235.982
f_dl14 <- forecast(mod_dl14, 
                       x= t(nc_data_test[,5]),
                       h=14,
                   interval = TRUE)
rmse(nc_data_test$log_cases,
     f_dl14$forecasts[,2])
## [1] 0.6221004
mae(nc_data_test$log_cases,
    f_dl14$forecasts[,2])
## [1] 0.5919259
checkresiduals(mod_dl14)
##            1            2            3            4            5            6 
##  1.536593406  1.551576009  1.503886164  1.486720187  1.607319380  1.525131019 
##            7            8            9           10           11           12 
##  1.401871490  1.412649754  1.350963648  1.355959990  1.374280435  1.375336126 
##           13           14           15           16           17           18 
##  1.340270000  1.406036663  1.407713604  1.395290258  1.353142299  1.327065848 
##           19           20           21           22           23           24 
##  1.335006793  1.298236744  1.107782248  1.068069622  1.070411373  1.048108430 
##           25           26           27           28           29           30 
##  0.997259244  1.000313879  0.963324283  0.890637343  0.886594946  0.879153095 
##           31           32           33           34           35           36 
##  0.872090716  0.927183748  0.949579678  0.890916913  0.874035580  0.869326150 
##           37           38           39           40           41           42 
##  0.827833339  0.757023153  0.545059025  0.738942774  0.716320886  0.614672632 
##           43           44           45           46           47           48 
##  0.503222798  0.524409773  0.525588834  0.525940541  0.415667627  0.355547982 
##           49           50           51           52           53           54 
##  0.263412225  0.328463759  0.284569382  0.094664284  0.095015991  0.368823153 
##           55           56           57           58           59           60 
##  0.392589306  0.374067781  0.288208620  0.311857977  0.468315526  0.493547057 
##           61           62           63           64           65           66 
##  0.292468184  0.207137041  0.110592804  0.221886199  0.257645419  0.071699049 
##           67           68           69           70           71           72 
##  0.033876328  0.351331202  0.358240280  0.333054258  0.307085786  0.111480896 
##           73           74           75           76           77           78 
##  0.115904691  0.116256398  0.115836320  0.045121171  0.029678459  0.018584964 
##           79           80           81           82           83           84 
##  0.215569948  0.219993742  0.220345449  0.288512124  0.333705503  0.312398251 
##           85           86           87           88           89           90 
##  0.381242644  0.429471028  0.433894823  0.434246530  0.455319782  0.400956742 
##           91           92           93           94           95           96 
##  0.311009526  0.321551561  0.319133319  0.323557114  0.323908821  0.323585160 
##           97           98           99          100          101          102 
##  0.327923813  0.321809543  0.324294188  0.341185259  0.345609054  0.345960760 
##          103          104          105          106          107          108 
##  0.313663019  0.327830253  0.271487922  0.200742797  0.187233589  0.191657384 
##          109          110          111          112          113          114 
##  0.192009090  0.199461340  0.082839121  0.090732513  0.071696075  0.007091785 
##          115          116          117          118          119          120 
##  0.011515580  0.011867287 -0.084691182 -0.077166288 -0.248023504 -0.231937149 
##          121          122          123          124          125          126 
## -0.243348866 -0.238925071 -0.238573364 -0.343822820 -0.427688526 -0.609362545 
##          127          128          129          130          131          132 
## -0.662759219 -0.704637034 -0.700213239 -0.699861532 -1.333506074 -0.787730300 
##          133          134          135          136          137          138 
## -0.749477427 -0.950486005 -0.989211645 -0.984787850 -0.984436143 -0.651405916 
##          139          140          141          142          143          144 
## -1.256919233 -1.257831917 -1.264666022 -1.211271437 -1.206847642 -1.206495935 
##          145          146          147          148          149          150 
## -1.188897781 -1.160548265 -1.205248274 -1.184074045 -1.326440063 -1.334438506 
##          151          152          153          154          155          156 
## -1.334183795 -1.545050378 -1.781865161 -1.836023690 -1.866861091 -1.732754075 
##          157          158          159          160          161          162 
## -1.701025221 -1.700721409 -1.443992009 -1.401718794 -1.443395606 -1.308954457 
##          163          164          165          166          167          168 
## -1.260139179 -1.292130475 -1.291672229 -1.799009565 -1.176305402 -1.179277836 
##          169          170          171          172          173          174 
## -1.155137562 -1.130080806 -1.123418340 -1.123008256 -0.392664066 -0.540048580 
##          175          176          177          178          179          180 
## -0.352125651 -0.183074443 -0.031884634 -0.033014202 -0.032881923  0.168705864 
##          181          182          183          184          185          186 
##  0.189706464  0.047979883  0.076721592  0.216238175  0.213594953  0.213310121 
##          187          188          189          190          191          192 
##  0.432565593  0.573412077  0.407552322  0.525367395  0.573217045  0.541321379 
##          193          194          195          196          197          198 
##  0.540842936  0.651728114  0.903001334  0.460022129  0.400370189  0.363782137 
##          199          200          201          202          203          204 
##  0.332934265  0.333058933  0.481871860  0.255649960  0.409269991  0.496957823 
##          205          206          207          208          209          210 
##  0.548900285  0.347292287  0.347665591  0.382024461  0.407945670  0.331239052 
##          211          212          213          214          215          216 
##  0.316304987  0.331429143  0.412483890  0.412541656  0.527292753  0.537463943 
##          217          218          219          220          221          222 
##  0.406004024  0.426740914  0.433951963  0.418395655  0.418603265  0.448635783 
##          223          224          225          226          227          228 
##  0.420814155  0.391068798  0.361120007  0.329769986  0.281579274  0.281930981 
##          229          230          231          232          233          234 
## -0.274733520  0.232373921  0.140850293  0.127591516  0.117234424  0.121658219 
##          235          236          237          238          239          240 
##  0.122009926  0.664667100  0.338270240  0.319242174  0.326261086  0.423635816 
##          241          242          243          244          245          246 
##  0.428059611  0.428411318  0.201242861  0.171557804  0.115374395  0.092602770 
##          247          248          249          250          251          252 
##  0.053013074  0.057436869  0.057788576  0.024598478 -0.023296574 -0.063682570 
##          253          254          255          256          257          258 
## -0.058111939 -0.096731358 -0.092307563 -0.091955857 -0.100665107 -0.150021215 
##          259          260          261          262          263          264 
## -0.229055132 -0.197738339 -0.197450836 -0.193027041 -0.192675335 -0.265602193 
##          265          266          267          268          269          270 
## -0.309131200 -0.389798693 -0.450531321 -0.476029792 -0.471605997 -0.471254290 
##          271          272          273          274          275          276 
## -0.559297759 -0.599962693 -0.654694245 -0.652054504 -0.719897776 -0.715473982 
##          277          278          279          280          281          282 
## -0.715122275 -0.692887335 -0.728766550 -0.798425504 -1.066043323 -0.821629934 
##          283          284          285          286          287          288 
## -0.817206140 -0.816854433 -0.891766642 -0.901684353 -0.978883482 -0.818635115 
##          289          290          291          292          293          294 
## -1.088197766 -1.083773972 -1.083422265 -0.982899384 -0.989307289 -0.998115463 
##          295          296          297          298          299          300 
## -1.171843391 -0.930450552 -0.926026757 -0.925675050 -1.014784302 -1.061162645 
##          301          302          303          304          305          306 
## -1.115629910 -0.895088156 -1.074820311 -1.070396516 -1.070044810 -0.965244462 
##          307          308          309          310          311          312 
## -1.040414230 -1.118084229 -1.379194720 -1.647778669 -1.643354874 -1.643003167 
##          313          314          315          316          317          318 
## -1.324028681 -1.289621514 -1.191160126 -0.880871535 -0.667306988 -0.662883193 
##          319          320          321          322          323          324 
## -0.662531486 -0.528429716 -0.478331805 -0.435727974 -0.370518493 -0.324786740 
##          325          326          327          328          329          330 
## -0.320362945 -0.320011239 -0.317096496 -0.359947445 -0.444457906 -0.439172623 
##          331          332          333          334          335          336 
## -0.409162705 -0.404738910 -0.404387203 -0.392650170 -0.403183856 -0.437775497 
##          337          338          339          340          341          342 
## -0.678159760 -0.940794454 -0.936370659 -0.936018952 -1.811231256 -0.185814212 
##          343          344          345          346          347          348 
## -0.343343114 -0.000848807  0.013831312  0.018255107  0.018606814  0.772053143 
##          349          350          351          352          353          354 
##  0.476651120  0.732228594  0.828306109  1.066478690  1.070902485  1.071254192 
##          355          356          357          358          359          360 
##  1.206353485  1.250855746  1.226596184  1.367906100  1.436611294  1.441035089 
##          361          362          363          364          365          366 
##  1.441386795  1.060259803  1.709445524  1.676270617  1.630225501  1.476468974 
##          367          368          369          370          371          372 
##  1.480892769  1.481244476  2.037238073  1.574126100  1.507732544  1.596623841 
##          373          374          375          376          377          378 
##  1.748396762  1.697942945  1.697866152  1.429741633  1.447093373  1.150974958 
##          379          380          381          382          383          384 
##  0.977041114  0.879864019  1.027309129  1.027624127  0.671654076  0.499273671 
##          385          386          387          388          389          390 
##  0.639327250  0.521937025  0.478084054  0.304646360  0.305806787  0.249435775 
##          391          392          393          394          395          396 
##  0.166801788  0.355509520  0.430344089  0.366283556  0.429895045  0.430862451 
##          397          398          399          400          401          402 
##  0.384507750  0.263184454  0.191789518  0.030870869 -0.019281431  0.150064647 
##          403          404          405          406          407          408 
##  0.150983831 -0.194235131 -0.247649753 -0.171137346 -0.193966235 -0.250361891 
##          409          410          411          412          413          414 
## -0.232545257 -0.231562172 -0.219525362 -0.275266216 -0.135565791 -0.171699933 
##          415          416          417          418          419          420 
## -0.088964302  0.028150138  0.029007791 -0.061154063 -0.174078710 -0.095048432 
##          421          422          423          424          425          426 
## -0.130341489 -0.152091940 -0.098790597 -0.098074107  0.023471879  0.047418264 
##          427          428          429          430          431          432 
##  0.161670229  0.298407458  0.393178738  0.469642037  0.470159878  0.575000338 
##          433          434          435          436          437          438 
##  0.529072399  0.593047211  0.605653855  0.518937020  0.538345220  0.538638850 
##          439          440          441          442          443          444 
##  0.455571381  0.421132727  0.130417922 -0.207275816 -0.525974878 -0.529672662 
##          445          446          447          448          449          450 
## -0.529829378 -1.420394844 -2.290911786 -0.399258326 -0.445433336 -0.457853386 
##          451          452          453          454          455          456 
## -0.445247845 -0.445610524 -0.462559236 -0.446234905 -0.638056496 -0.682071998 
##          457          458          459          460          461          462 
## -0.718082998 -0.800597143 -0.800693401 -0.824778925 -0.836709246 -0.387741136 
##          463          464          465          466          467          468 
## -0.404803053 -0.425674312 -0.507977759 -0.507763137 -0.506707833 -0.530815066 
##          469          470          471          472          473          474 
## -0.573940919 -0.569819727 -0.567717078 -0.625579768 -0.625370813 -0.583592938 
##          475          476 
## -0.581894113 -0.811369602

nc_data_test <- cbind(nc_data_test,f_dl14$forecasts[,2],
                      f_dl14$forecasts[,1],
                      f_dl14$forecasts[,3])

png(filename = "sensitivity_nc_dl.png",res = 700, units = "cm",
    width = 20, height = 10)
dl_plot <- nc_data_train %>% ggplot(aes(date,log_cases)) + 
  geom_line() + 
  geom_ribbon(data = nc_data_test , aes(ymin = f_dl14$forecasts[,1], 
                                        ymax = f_dl14$forecasts[,3]),
              fill = adjustcolor( "red", alpha.f = 0.2)) + 
  geom_line(data = nc_data_test,aes(date,log_cases,color="Actual")) +
  geom_line(data = nc_data_test,aes(date,f_dl14$forecasts[,2],color="Forecasted")) +
  scale_colour_manual(values=c("Actual"="cyan", "Forecasted"="red"), 
                      labels=c("Actual", "Forecasted")) +
  theme_bw() + theme(legend.position = "none") + ylab("")
dev.off()
## quartz_off_screen 
##                 2
exp(f_dl14 $forecasts[1,2])
## [1] 26.53398
exp(f_dl14  $forecasts[1,1])
## [1] 5.364831
exp(f_dl14 $forecasts[1,3])
## [1] 141.1164
exp(f_dl14  $forecasts[1,2]) - exp(nc_data_test[1,4])
## [1] 14.85072
exp(f_dl14  $forecasts[7,2])
## [1] 28.45966
exp(f_dl14  $forecasts[7,1])
## [1] 5.764687
exp(f_dl14  $forecasts[7,3])
## [1] 148.9824
exp(f_dl14 $forecasts[7,2]) - exp(nc_data_test[7,4])
## [1] 10.31143
exp(f_dl14  $forecasts[14,2])
## [1] 31.09752
exp(f_dl14 $forecasts[14,1])
## [1] 7.046847
exp(f_dl14 $forecasts[14,3])
## [1] 169.9599
exp(f_dl14  $forecasts[14,2]) - exp(nc_data_test[14,4])
## [1] 5.768146
#forecasting

png(filename = "sensitivity_plots.png", res = 700,
    units = "cm", width = 20, height = 27)
grid.arrange(arima_plot,
             sarima_plot,
             arimax_plot,
             sarimax_plot,
             dl_plot,
             ardl_plot,
             ncol=1,
             left = text_grob("Logarithm of New COVID-19 cases per 10K", rot = 90, vjust = 1))
dev.off()
## quartz_off_screen 
##                 2